m-ISOMETRIC TRANSFORMATIONS OF HILBERT SPACE, I
Jim Agler and Mark Stankus
Department of Mathematics, Univ. Of Calif., San Diego
La Jolla, CA 92093-0112, U.S.A.
m
A model for operators satisfying the equation
2: (_l)k(';:)(T*)m-k
T m- k = 0
k=O
is given as multiplication by e i ¢ on a Hilbert space whose inner product is defined in terms
of periodic distributions and we relate this model theory for the case when m = 2 to a
disconjugacy theory for a subclass of Toeplitz operators of the type studied by Boutet de
Monvel and Guilliman, classical function theoretic ideas on the Dirichlet space, and the
theory of nonstationary stochastic processes. This is presented in a series of three papers.
In this first paper, we concentrate on a model for these T.
Table of Contents
Paper I.
O.
1.
2.
3.
4.
Introduction
The Elementary Operator Theory of m-Isometries
Distribution Differential Operators on the circle
A model for m-isometries
Smoothing applied to m-isometries
Paper II.
Introduction to Paper II
5. Brownian shifts, Brownian unitaries and Brownian isometries
6. A Disconjugacy Theorem for Matricial Toeplitz Operators
7. Another proof of the Lifting Theorem
Paper III.
Introduction to Paper III
8. A cyclic vector
9. Brownian Unitaries Revisited
10. Some remarks and Open Questions
~0. I n t r o d u c t i o n
In this paper we shall study the bounded linear transformations T of a complex Hilbert
space 7-/that satisfy an identity of the form
1
for some positive integer 'm. Operators T satisfying (0.1) are said to be m-isometrics.
The case when m = 1 has already received definitive analysis by operator theorists, a
circumstance due in large part to the fundamental importance of an adequate theory
of isometrics to the intrinsic problem of modeling the general contractive operator via
its isometric dilation ( [ B r - R ] , [Sz.-N-F]) as well as to the development of a successful
L 2 theory for stationary stochastic processes, to the Hoo-control problem [H3], and to
the function theory of the classical Hardy space [B,Hel]. To develop an analysis of misometrics in the case when m > 1 requires ideas from many areas of mathematics. In
particular, in this paper we shall employ the theory of periodic distributions to derive a
function theory model for m-isometrics, a disconjugacy theory for a subclass of Toeplitz
operators of the type studied by Boutet de Monvel and Guillemin [M-G] and function
theoretic ideas on the classical Dirichlet space. In addition, operators arising from a
certain class of nonstationary stochastic processes related to Brownian motion will play
an essential role in the theory of 2-isometrics.
Our basic approach will be based on two ideas of J. W. Helton who in [HI] initiated
the study of operators T which satisfy an identity of the form
(0.2)
T * ' ~ - ( m1) T*'~-~T + . . . + ( _ I ) ~ T m = O.
Helton's first idea was that such operators possess a model as multiplication operators on
Hilbert spaces of C~176
completed with respect to a inner product
(0.3)
[~, +] = [
]
where L is a positive definite differential operator on a compact interval I c R. Tile second
idea [H2] was that the general operator T satisfying (0.2) can be explicitly represented in
the form
T = J IZ4
where J is a real jordan operator (i.e., J = A + Q, A = A*, Q A = AQ, and Q is nilpotent)
and that such a representation is formally equivalent to the classical disconjugacy theory for
elliptic ODE. In this paper we shall carry out Helton's first idea for the case of m-isometrics.
This results in a general model for the finitely cyclic m-isometry as multiplication by e ~~
on the completion of a pre~Hilbert space of analytic functions with respect, to an inner
p r o d u c t as in (0.3) where L is a differential o p e r a t o r on 0D whose compression to the
analytic C ~ - f u n c t i o n s is positive definite. In addition we shall carry out Helton's second
idea for the case m = 2. This results in a theorem t h a t represents the general 2-isometry
T in the form
T= BIAd
where B has the block form
(0.4)
0
a > 0 is a constant, V is an isometry, U is u n i t a r y and E is a Hilbert space isomorphism
onto ker V*. T h e o p e r a t o r s B of the form in (0.4) are referred to as Brownian unitaries of
covariance a a n d were discovered by considering certain n a t u r a l examples of 2-isometrics
arising from statistics.
T h e p a p e r is divided into ten sections and is published in three parts. T h e first p a r t
consists of Sections 1-4 and deals with the general theory of m-isometrics, tile second p a r t
consists of Sections 5-7 and prove a spatial lifting theorem for 2-isometrics. and the third
p a r t consists of Sections 8-10. Sections 8 and 9 analyze cyclic 2-isometrics in more detail
and s t a t e a n u m b e r of open questions related to rn-isometries. A l t h o u g h the p a p e r is quite
long the reader should not conclude t h a t the theory of m-isometrics (even in the case when
m = 2) is any way complete. In Section 10 we discuss a number of unsolved problems and
possible directions for further research.
In Section 1 we discuss the elementary o p e r a t o r theory of m-isometrics. It turns out
t h a t if T is an m - i s o m e t r y and A T is defined by
AT = (r*)m-l . rm-l _ (m-1)(r*)m-2rrn-2
1
+ ... + ( _ l ) m - i
then A T is a positive operator, kernel AT invariant for T and T ! ker/k T is an m -- 1 - isometry. These facts can be milked for a considerable amount of information. In particular,
several notions of p u r i t y are introduced and several canonical block m a t r i x representations
for m-isometrics are presented. In addition, if m = 2 and T is finitely cyclic, t h e n AT is
c o m p a c t (c.f. [P] [ B e t - S ] ) .
T h e spectral picture of an m - i s o m e t r y is easy to describe. If T is an m-isometry, then
aap(T) C_ o ~ and so a ( T ) C_ 0D or a ( T ) = D -
(Lemma 1.21). Consequently, ale(T) C_ 0D
and T - A is semi-Predholm for each A C II) and in particular,
ind(T-
A) - ind T < 0
for all A E D.
In p a r t i c u l a r there is the following spectral dichotomy: if T is invertible, t h e n a ( T ) C_
0tD; if T is not invertible, then a ( T ) = D - .
In b o t h cases it turns out t h a t there is a
functional calculus (Proposition 1.20). In the invertible case the functional calculus is a
continuous homomorphism of 79, the algebra of C ~ functions defined on the unit circle, and
in the noninvertible case the calculus is a continuous homomorphism on/3~, the algebra
of analytic functions on the disc with C ~ boundary values.
In Section 2 we introduce the class of distribution differential operators (DDO). A
DDO L is a linear mapping from 79 into/?~, the space of periodic distributions, that has
the particular form
for some/~o, � 9 � 9 fin E / ? ' . In (0.5), we follow the usual convention for defining the product
of an element in 79 by an element in :D' by defining ~(I)/~l E D' by ~(l)r162 = ~z(~(t)r
Thus, a DDO is simply a differential operator on the circle with distribution coefficients
and if in (0.5) the coefficients ~t are functions, then L is a classical differential operator.
The significance of DDO to the theory of rn-isometries is revealed in Section 3. It
turns out that if T is a m-isometry with cyclic vector % then there exists a unique DDO
L with the property that
(0.6)
(~(T)% ~(T)~) = L(~)(~)
for all ~, r e :D~. In (0.6) ~(T) and r
are formed using the functional calculus. (0.6)
then allows one to construct a model for the general bicyclic invertible m-isometry as multiplication by e i~ on the form domain of a DDO (Theorem 3.14), and in the noninvertible
case to construct a model for the general cyclic m-isometry as multiplication by e i~ on the
analytic form domain of a DDO (Theorem 3.23). In the noncyclic cases models are obtainable by considering DDO with matrix coefficients (Theorems 3.44 and 3.49). Finally,
the problem of how the DDO L in (0.6) transforms under changes of cyclic vector ~/ is
considered in Theorem 3.52.
Translation and convolution can be extended in a canonical way to DDO and these
operations on the DDO L that represent m-isometries T via (0.6) correspond to interesting
Hilbert space constructions on the operator T. In particular through the use of an approximate identity the L attached to a general m-isometry via (0.6) can be smoothed and
some questions about m-isometries reduced to the case when L is a classical differential
operator. This technique was first used in [Agl] to settle the lifting conjecture of Helton
concerning 3-symmetric operators (equation (0.2) with m = 3) and in the present context
is applied in Section 7 to establish the equivalence of a disconjugacy theory for classical
analytic DDO of order 1 (Lemmas 6.7 and 6.16 and Proposition 6.17) and an algebraic
lifting theorem for 2-isometries (Theorem 5.80).
Papers II and III of this series are devoted to the study of 2-isometries. In Section
5 a certain class of 2-isometries motivated by statistical considerations is introduced, the
Brownian unitaries. The irreducible Brownian unitaries correspond to the time shift operator on a scaled Brownian motion process and are referred to as Brownian shifts (Definition
5.5). Each Brownian shift is determined by a nonrandom rotation angle 0 and a positive
covariance scalar a. It turns out that there is a direct integral theory for Brownian unitaries (Theorem 5.20), each is a direct integral with multiplicity over angles of Brownian
shifts of a fixed covariance a. The scalar a is referred to as the covariance of the Brownian
unitary. F~arthermore, the Brownian shifts in addition to their appearance as the time
shift for perhaps the most ubiquitous of nonstationary Ganssian stochastic processes have
elegant characterizations on the level of operator theory (Proposition 5.6) as well as within
the field of DDO (Proposition 5.2). The principal result of Section 5, the Lifting Theorem
for 2-Isometries (Theorem 5.80), asserts that the general 2-isometry T with IIATII _< a 2
can be represented as the restriction to an invariant subspace of a Brownian unitary of
covariance a.
The proof of the Lifting Theorem in Section 5 consists of a synthesis of a proof of
the Sz.-Nagy Dilation Theorem and a proof of the familiar fact that isometries possess
extensions to unitaries, and in particular, is purely algebraic in character. The original
proof of the Lifting Theorem was analytic in character and derives from first principles
the class of objects the 2-isometries are to be lifted to (i.e., the Brownian unitaries). The
main analytic content here is a discongugacy theory for regular first order DDO with L ~176
coefficients discussed in Section 6 (Lemmas 6.7 and 6.16 and Proposition 6.17).
A second class of operators discussed at length in Section 5 is the collection of direct
integrals with multiplicity in 0 of Brownian shifts with varying covariance which we refer
to as Brownian isometries. This class not only arises naturally in Theorem 6.20 which
studies the extensions of 2-isometries defined as multiplication by e w on the form domain
and/30 = w 27r
dO with w c L ~ ) but
of Toeplitz operators (i.e., (0.5) with m = 1,/31 : ~dO
,
also arises naturally in the general theory of Brownian unitary extensions which is studied
in Section 9. Both the classes of Brownian unitaries and Brownian isometries possess
elegant C*-algebraic characterizations (Theorems 5.20 and 5.48).
Sections 8 and 9 are devoted to the study of cyclic 2-isometries. It turns out that if
T is a cyclic 2-isometry, then
din, ker ( l l A w l l - AT) = 1
and furthermore if f0 ~ 0 is chosen so that AT f0 = IlATllf0, then f0 is cyclic for T if
and only if T is pure (Theorem 8.19). Here, a 2-isometry T is said to be pure if it has no
isometric direct summand. If one chooses 7 = f0, then the DDO L attached to T via (0.6)
turns out to have a particularly nice form. This allows one to construct a model for T
based on a probability measure on the unit circle (Theorem 8.32). Finally, Section 9 uses
Theorem 8.19 to derive a number of qualitative facts about the existence and uniqueness
of Brownian unitary and isometric extensions of a cyclic 2-isometry.
The class of 2-isometries has also been independently studied by Stefan Richter [R1],
[R2]. His work develops a Dirichlet space type model for 2-isometrie~s by exploiting a
beautiful extension of the wandering subspace argument for isometries to the class of 2isometrics type which has considerable overlap with Section 8 of this paper.
This model leads to a rich function theoretic analysis of 2-isometrics.
~1. The Elementary Operator Theory of m-Isometrics
In this section, we shall introduce much of the notation of the paper and give some
basic properties of m-isometrics. We shall then specialize to the case m = 2 and obtain
additional information.
All Hilbert spaces shall be complex and separable. If ~ is a Hilbert space we let s
denote the bounded linear transformations of 7-/. If T E s
we define ( y x - 1)re(T) E s
by the formula,
and m is a positive integer
k=0
If T E s
and ( y x - 1)re(T) = 0, then we shall say that T is an m - i s o m e t r y . A basic
fact in the theory of m-isometries is that if T ~ s
is an m-isometry and AJ c_ 7-t is
an invariant subspace for T, then T[ AJ is an m~isometry. This follows from the algebraic
formula,
(1.1)
( y x - 1)e(TlAd) = P M ( y z -
1)e(T) I 2M
where T is an arbitrary operator on 7-/, 3.4 c_ 7-/ is invariant tbr T, PM denotes the
orthogonal projection of H onto M , and ~ is a nonnegative integer.
A key object in the theory of ru-isometries is the s y m b o l o f T, sT', defined for an
arbitrary T E s
by the equation
sT(k) = T*kT k
and for an invertible T E s
(1.2)
Computations involving
series for ST(k),
(1,3)
,
k>_0..
,
k G Z.
by the equation
sT(k) = T * k T k
ST(k) c o m e
out quite clearly if we consider the formal difference
sT'(k) = ~
f=0
f3e(T)k (t/,
where k (~) = k. (k - 1 ) . - - ( k -
g + 1) and
1
~9~(T) = ~
(yx
-
1)e(T).
If a = {ak} is a sequence of scalars or operators we define a sequence /)a by setting
6a = {ak+l -- a~}. With this notation 5{k (e)} = (k _> 0), and formula (1.3) is derived by
applying 5e to (1.2) and setting k = 0.
Now observe that if T E s
and fie(T) = 0 then ~j(T) = 0 whenever j _> L Hence
if T E s
one sees via (1.3) that T is an m-isometry if and only if ST(k) is a polynomial
in k of degree less than m. A deeper fact that also follows from (1.3) concerns the object
of the following definition.
D e f i n i t i o n 1.4. If T E L:(7~) is an m-isometry, then AT, the covariance operator o f T , is
defined by
AT = flm-l(T)
�9
Note in the definition of A T we have in our notation suppressed the dependence of A T on
rr[.
The following propositions illustrate the interest of AT.
P r o p o s i t i o n 1.5. Is T is an m-isometry, then /kT k O.
PROOF.
The proof consists of the following computation.
{ A T X , X } = lim
k~
= lira
k~oc
1
k(m-l) (ST(k)x,r
1
k (m-l)
iiTkxll
>_0.
P r o p o s i t i o n 1.6. /f T C /:;(7-/) is an m-isometry, then ker(AT) is invariant for T and
T I ker AT is an m -- 1 isometry. Fbrthermore, if M C 7-I is invariant for T and T ] A,/ is
an ( m - 1)-isometry, then 3.4 C_ ker AT.
PROOF.
S i n c e T is a n
m-isometry,
T * A T T - A T =/3re(T) =- 0
and in addition Proposition 1.5 implies that
AT>_O.
To see that ker(GT) is invariant for T observe that if x c ker AT, then
( A T T z , T . ) = (T*ATTx, z)
:~)
~0,
That in fact T] ker AT is an (m--1)-isometry follows from (1.1). Finally, to see that
M C_ ker A r whenever 54 C_ ~, TAd c_ A4 and T ] 3 4 is an ( m - 1 ) - i s o m e t r y fix x E M .
Using (1.1) we see that
(/~r x, :?g)= (~m l(Z)x, ,2:}
= (/~,~-,(r 154)*, :~:)
= 0.
Hence via Proposition 1.5 we conclude that x E ker AT.
This completes the proof of
Proposition 1.6.
Note that an application of Zorn's Lemma gives that any operator possesses a maxinml invariant subspace 34 (possibly equal to {0}) such that T t 5 4 is an , , , - 1-isometry.
Proposition 1.6 guarantees that if T is an m-isometry, then such an 34 is unique (in fact,
equal to ker AT).
In the next proposition we examine maximal reducing subspaces on which a given
m-isometry is r n - 1-isometric.
Proposition
1.7. / f T C s
is an m-isometry, then there exists a unique subspacc
34 C_ 7-[ that is maximal with respect to the following properties:
(i) 34 is reducing for T, and
(ii) T I 34 is an ( ~ - 1)-isometry.
PROOF.
The existence of subspaces M that are maximal with respect to (i) and (ii)
follows immediately from Zorn's Lemma. To prove uniqueness it suffices to establish that
if 541 and 342 satisfy (i) and (ii), then so also 34 = (541 + 3.42)- satisfies (i) and (ii).
That 34 satisfies (i) is immediate. To see that 34 satisfies (ii) first observe that since
T I 541 and T [ 542 are ( m - 1)-isometries Proposition 1.6 implies that
A41_Cker AT
and
342Cker
AT.
Hence by another application of Proposition 1.6,
34 = ( 3 4 1 + A 4 2 )
C_ker A T ,
and (ii) holds for 54. This concludes the proof of Proposition 1.7.
Proposition 1.7 motivates the following definition.
be an m-isometry and g a positive integer. T is g-pure if
D e f i n i t i o n 1.8. Let T E s
T has no nonzero direct s u m m a n d which is an g-isometry. T is pure (as an rn-isometry) if
T is ( m - 1)-pure.
We remark before continuing that if T is an m-isometry and not an ( m - 1)-isometry,
then T may or may not be pure as an rn-isometry but is never pure as an ( r n + 1)-isometry.
Also note that if T is pure as an rn-isometry, then T is not an ( r n - 1 ) - i s o m e t r y and in
addition, T is g-pure for all g < r n - 1 . Finally, note that the absence of 0-isometrics on
spaces of positive dimension implies that all l-isometrics are pure, conflicting with the
usual use of the term pure in the context of isometrics.
We now use Proposition 1.7 to derive a canonical decomposition of an rn-isometry into
a direct sum of pure g-isometrics, g _< m. Fix an rn-isometry T E s
By Proposition
1.7, there exists a decomposition H = (H @ Hm) @ ~,~ with the property that if jr4 C_ H
is reducing for T and T I Ad is an ( r n - 1)-isometry, then Ad C_ H @ Hm. In particular,
(1.9)
(1.10)
H,~
T I~,~
is reducing for
is a pure r n - isometry
T,
(if
7t,~ r {0})
and
(1.11)
T I ~ @ 7-&
is an ( m - l ) - isometry.
Now repeat the application of Proposition 1.7 with T replaced by T I H G 7ira to obtain a
space H m - 1 C_ "]~ G ~'~rn such that
(1.12)
(1.13)
7-&-1
TIH.~-I
is reducing for
isapure(m-1)-isometry
T,
(if
7-/m-1 r
and
(1.14)
r l ~ ~ (~-~m-1 @ ~-~rn)
is a n (~t -- 2) --
isometry.
Continuing in this way we obtain a decomposition,
(1.15)
with the properties that
(1.16)
~e
is reducing for
T, 1 < g < r n ,
and
(1.17)
T[He
isapure
f-isometry,
1 <f<m.
Furthermore, it is easy to see that by the construction that any decomposition satisfying
(1.15), (1.16), and (1.17) is unique. In particular, one sees that if T is an m-isometry and
f _< m - 1, then there exists a unique maximal reducing subspace ,M c H such that T I M
is an f-isometry (Ad = ~1 |
�9 ~/e).
On a pragmatic level the decomposition of (1.15) is of value in that most operator theoretic questions about m-isometrics will reduce down to questions about pure g-isometrics
for ~_< m.
A construction similar to that just implemented for reducing subspaces using Proposition 1.17 can be carried out for invariant subspaces using Proposition 1.6. This results
in a decomposition,
(1.18)
7-/= 7-tl |
| 7-/m
where M e = 7-/1 | ..- | He is invariant for T, T I Ade is an g-isometry and Me is maximal
with respect to these properties.
If T is an m-isometry it turns out that T possesses a C~-functional calculus. Let 79
denote the Frechet algebra of C ~ functions on the circle and define :Da C ~9 by setting
D~,.={~ET?:~(k)=0
whenever
k<0}.
We say that T E s
has a Da-functional calculus if there exists a continuous unital
algebra homomorphism 7r : :D~ ~/2(7-/) such that ~r(1) = 1 and ~r(ei~ = T. If, in addition,
T is invertible we say T has a ~?-functional calculus if there exists a continuous unital
algebra homomorphism 7r : 7? -~ s
such 7r(1) = 1 and 7r(ei~ = T. To see that an
m-isometry T E s
does indeed have a Da functional calculus, define ~r : 9�
(1.19)
~(~) = ~
-~ s
by
~ (k)r k
k=0
To see that (1.19) in fact defines a continuous algebra homomorphism it suffices to check
that 7r is well defined and continuous. Note that the topology on Z)a is given by the
seminorms
1
II~lla
(k + 1)2~1~^ (k)12 ~
T/'t
,
Also observe that if ~ > ~- and T is an m-isometry, then
oc
k
0
(~>
O.
since ST(k) is a polynomial of degree <_ m - 1. Thus, if %oE ~Da and c~ > ~-,
O<3
< ~ Iw^(k)l IITkll
k~=o~ (k)Tk
k=0
= ~ (k~l~^(k)l)(k-~llTkll)
k=O
so t h a t :r is continuous. It is equally easy to define the analogous h o m o m o r p h i s m o f / 9 in
the case when T is invertible. Summarizing we have established the following proposition.
1.20. Let T E s
be an m-isometry. T has a lPa-functional calculus. If
T is invertible, then T has a 1P-functional calculus.
Proposition
We now describe the spectral picture of an m-isometry. If T is an m-isometry, then
ST(k) is a p o l y n o m i a l in k and so the spectral radius formula implies t h a t a ( T ) C D - .
The following l e m m a gives a more detailed description of the spectrum.
Lemma
1.21. / f T E s
is an m-isometry, then the approximate p o i n t spectrum lies
in the unit circle. Thus, either a(T) COD or a(T) = D - . In particular, T is injective and
r a n ( T ) is closed.
PROOF.
If A E C is in the a p p r o x i m a t e point s p e c t r u m of T, then there exist unit vectors
xt ~ 7Y such t h a t (T - A)x~ --~ 0. By induction, for each integer k > 1, (T k - Ak)xe ~ O.
Thus
m
) /
kk=O
k=O
k=O
= (IAI2 - 1) m
and so IAI = 1. Since Oo'(T) c O'ap(T) c_ OlD, r
c_ cgD or a ( T ) = D - . Since 0 r cr.p(T),
T is injective and r a n ( T ) is closed. This establishes L e m m a 1.21.
Before continuing, we make the following remarks concerning the spectral picture of
an m - i s o m e t r y T.
Since
F i x A E D.
Crp(T) C_ OlD, the
Since ae~(T) C_ a~p(T) C_ OD, T - ;~ is semi-Fredholm.
Fredholm index of T - ~, i n d ( T - A), is nonpositive. Finally, since
ace(T) C 0D, the function from D into Z U {+oc} given by ~ ~ i n d ( T
A) is constant
(See, for example,[C]).
Corollary 1.22. Le~ T E s
PROOF.
/ / b o t h T and T* are m-isometrics, then ~(T) c_ OlD.
We argue by contradiction. By L e m m a 1.21, if a(T) ~ 0ID, then a ( T ) = D .
Since L e m m a 1.21 also implies t h a t 0 r Crop(T), we see t h a t ( r a n T ) - # 7-/. Hence 0 E
trap(T* ) contradicting L e m m a 1.21 and the fact t h a t T* is an m-isometry.
Thus, there are two cases: if T is an m-isometry either T is invertible and a ( T ) C_ 0D
or T is not invertible and a ( T ) = D - . In fact b o t h of these cases cannot occur for all m.
Proposition
1.23. If T is an invertible m-isometry and m is even, then T is an ( m - 1 ) -
isometry.
PROOF.
If T is invertible, then one deduces as in the proof of Proposition 1.5 (let
k -* - o c ) t h a t - - A T > 0. Hence A T = 0, i.e. T is an ( r n - 1 ) - i s o m e t r y .
Note t h a t Proposition 1.23 implies t h a t every invertible 2-isometry is a u n i t a r y operator.
W h e n applying the above observations to the case of 2-isometries, one gains additional
structure. If ~t is a Hilbert space and S is a subset of 7-t, then we define VS to be the
smallest subspace of 7-I which contains S (i.e. VS is the closed linear span of S). T E/2(7/)
is finitely cyclic if there exist an integer n and there exist vectors ~1,- �9�9, 7n E 7-I such t h a t
V{Tk~/j : k > 0, 1 < j < n} = 7-{.
Proposition
PROOF.
1.24. / f T E s
is a finitely cyclic 2-isometry, then A T is compact.
Since T is finitely cyclic, T k is finitely cyclic and so ran(Tk) • is finite dimensional
for every k _> 1. In particular if P~ denotes the orthogonal projection of 7-/onto ran(Tk),
then / ~ T -- P k A T P k is finite rank. Now fix x E 7-I and let y = Tkx. Since T is a 2-isometry,
T*Z~TT = A T , and thus
(PkATPky, y) = (ATy, y)
= (ATTkZ, Tkz)
= (T*kATTkX, Z)
= (Z~TX, X).
On the other hand, (1.3) implies that
1
(ATZ, z) = ~. (IIT%II 2 -Ilxlt 2)
Hence
1
(P~ATPky, y} = g (IIT%II 2 -ilxll 2)
1
-< K IITkxll 2
1
= -llyll
k
a n d w e see t h a t I I P k A T P k l l
2
---* O. H e n c e
A T =
lim ( A T -- P k ~ T P k )
k-~oo
is compact. This concludes the proof of Proposition 1.24.
In the case of 2-isometrics Proposition 1.7 can be improved upon in that the subspace
Ad of that proposition can be concretely identified.
P r o p o s i t i o n 1.25. Let T E f-.(~) be a 2-isometry and 3 / be the smallest invariant
subspace for T generated by (ran A T ) - . 3 / reduces T and 3/• is the largest reducing
subspace on which T is isometric.
PROOF. To show that N reduces T, it suffices to show that r a n ( T ' T h A T ) c 3 / f o r all
n=0,1,2 .....
By Proposition 1.6, ker AT is invariant for T and so ( r a n A T ) - is invariant for T*.
Thus r a n ( T ' A T ) C_ ( r a n A T ) - C_3/.
If x E H and n > 0, then
T*(TnATx) = T*T(Tn-IATx)
= (AT + I) (T~-IATx)
E ranAT § ranT ~- 1AT
c_3/.
Thus 3 / r e d u c e s T.
Suppose now that 3/o reduces T and T 13/0 is isometric. By Proposition 1.6, 3/o c
ker AT and so 3/oz _D ranAT. Since 3/0z is invariant for T, 3/oz _D 3/. Thus No C 3/•
This establishes Proposition 1.25.
We close this section with a decomposition theorem for 2-isometrics that contains
both the decompositions of (1.15) and (1.18).
T h e o r e m 1.26. L e t T E s
be a 2 - i s o m e t r y . I f Ad is t h e largest r e d u c i n g s u b s p a c e for
T on w h i c h T is i s o m e t r i c t h e n T h a s t h e b l o c k r e p r e s e n t a t i o n
T=
(1.27)
i00 V0 o 1
ko
0
with respect to the decomposition,
(1.28)
= Ad �9 (ker /~T O J~) ~ ( r a n A T ) - .
I51rthermore,
(1.29)
V0 is an isometry,
(1.30)
V
is a pure isometry,
(1.31)
(1.32)
V'E=0,
X*(E*E
(1.33)
+ X*X
ran E
- 1)X = E * E + X * X
is dense in
- 1,
kerV*,
and
(1.34)
ker(E*E + X * X - 1) = {0}.
PROOF. By Proposition 1.6, k e r A t is invariant for T and Ad s kerAT. Hence T has
tile form of (1.27) with respect to the decomposition (1.28) and (1.29) holds. Computing
using (1.27) one finds that
A T =
V*V
1
E*V
-
V*E
E*E + X*X
.
- 1
Hence V is an isometry, and (1.31) and (1.34) hold. Computing
T* A T T
= AT"
yields that (1.32) holds. Now observe that Proposition 1.25 implies that
(1.35)
v { y k r a n E : k >_ 0}
is dense in
ker AT OA/t.
Since V is an isometry and ran E is in the initial space of V (i.e. ran E c_ ker V*), (1.35)
immediately implies that V is pure and ran E is dense in ker V*. This concludes the proof
of Theorem 1.26.
We observe in Theorem 1.26 that if T is a 2-isometry that has the representation of
(1.27) but with respect to a decomposition
(1.36)
7-/= 7-lo | 7-ll q) 7-/2
and with (1.29)-(1.34) holding then necessarily 7-/0 = A4, the largest reducing subspace for
T such T I Jt4 is isometric, 7-ll = ker AT (9 A/l, and 7-12 = ( r a n A T ) - .
In the case of isometries one can do much better than Theorem 1.26 in that one can
through abstract arguments (IF]) explicitly display the general pure isometry as a direct
sum of cyclic pure isometries. That such an approach will not be successful for 2-isometrics
is seen by the following example of a noncyclic 2-cyclic pure 2-isometry:
0
0
0
S
0
0
0
1
0
~|
b
-1
~2. D i s t r i b u t i o n D i f f e r e n t i a l O p e r a t o r s on t h e circle
L e t / 9 = C ~176
(0D), the Frechet space of infinitely differentiable functions on the unit
circle. Let 79' denote the dual of :D, the space of distributions on the circle.
We define a linear operator D : i/) --~ ~) via the formula
D~
:
ld
~-~o.
If ~ > 0 is an integer, we define D (s) :/9 --*/P by
D (~) = D - ( D -
1).-. ( D - f + 1).
Recalling that eik~ E i/) is an eigenfunction for D with corresponding eigenvalue k, we see
that
D (t) (e ~k~ = k ( t ) e ikO '
a formula which we use repeatedly in the sequel.
Now recall that if fl E i/)' and ~a ff 19 then Ffl E / 9 ' can be defined by
~(~)
=/~(~),
~/~c l).
Thus, if/3 E 79~, /3 can be regarded naturally as a m a p / 3 : 7 9 ~ 79' by defining
/3(~) = ~/~.
W i t h the n o t a t i o n of the preceding two paragraphs, we are now able to define the
class of objects t h a t will be the subject of this section.
Definition
2.1.
A distribution differential operator DDO is a m a p L : 79 ~ 79~ t h a t has
the form
L = ~_2 /3tD(t)
(2.2)
/=0
w h e r e / 3 0 , . . . ,/3m E 79'. If L is as in (2.2) and/~m r 0, we say L has order m.
If L is a D D O and ~ , r E :D, let us agree to define C L ~ : D - , D ' by
('~L~) (X) = '(3L(qaX),
X E 79.
W i t h this definition observe t h a t if L is a DDO of order m and ~, ~ E D, then CLqo is
a DDO of order < m.
As a final piece of notation before stating our first lemma, we
introduce the formal derivative of a DDO.
Definition
2.3. If L = ~ / 3 e D (e) is a DDO of order m, we define the DDO d
L of order
m - 1 by setting
d L
dD
~-~g/3eD(e 1)
T h e following lemma is a key piece of algebra t h a t not only will lead to several characterizations of DDO, but will also allow us in Section 3 to characterize positive definite
DDO as the key analytical objects corresponding to the model theory of m-isometries.
Lemma
2.4. If L is a DDO, then
e-~~
d
i~ - L = ~ L.
PROOF. It suffices to establish the formula for the case L = flD (~). We observe t h a t
Leibnitz's rule is true for D (e) in the form:
j=0
\J/
Thus, using the fact t h a t DO)(e i~ = 0 if j > 1,
(e i~
i~
/3D(g))(~) = e i~176
= g/3D (e-~) (~).
This establishes L e m m a 2,4.
We o b t a i n the following immediate corollary.
+ gei~ (g 1)~) _/~(D(e)~)
Lemma
2.5. If L is a DDO and m > O, then
(2.6)
2.7. /f {akl,k~ } is bilateral, temperate and polynomial of degree m on diagonals, then there exists a unique DDO L of order m such that
Proposition
(2.8)
akl,k2 = L (e ik~~ (e ik2o) ,
k~, k2, E Z.
Conversely, if L is a DDO of order m and {ak~,k2} is defined as in (2.8), then {akl,k~ } is
bilateral, temperate and polynomial of degree m on diagonMs.
PaOOF. Let {ak~,k~ } be bilateral, temperate and polynomial of degree m on diagonals. If
k E Z is fixed, then aj,k+j is a polynomial in j of degree < m. Thus, there exist be,k E C
for g = 0 , 1 , . . . , m such that
aj,k+j
=
~
be,kj (e) ,
j E Z.
g=O
Since {akl,k2 } is temperate, we can define a distribution/3~ satisfying
(2.9)
/)e(k) = be,k.
Computing, we find that if L is defined by
L = ~
3~D (e) ,
g=0
then
=Z
g
g
= ~
be,kJ (t)
g
=
aj,k+j
�9
To see that L is unique, simply observe that if L = ~
g=0
holds, then (2.9) holds.
/3~D (e) is any DDO such that (2.8)
Conversely, let L be a DDO of order m and {akl,k~} be defined as in (2.8). For fixed
kl,k2 E g, we define a sequence {bj} by
bj = ah~+j,kz+j ,
j C N.
Note that L e m m a 2.5 implies that
( d
'
.
.
.
.
= 0
since L has order m. Hence {akl,k2} is polynomial of degree m. on diagona.ls. Finally, if
L = ~ 13tD(e), then
'~kl,k~ = E
f# ( (D(e)e'kl~
k.~o)
and so {akl,k2 } is temperate. This establishes Proposition 2.7.
The following corollary is the unilateral version of Proposition 2.7.
C o r o l l a r y 2.10. //{ak~,a: } is unilateral, temperate and polynomial or"de.~ree m on diagonals, then there exists a unique DDO L of order m such that
(2.11)
Conversely, i l l
O,k,:k2 = L(eiklO)
(C-ik20)
~:1:/~:2 ~ O.
is a D D O o / o r d e r m and {ak,,k~} is det/ned as in (2.11), ~hen {~Lk,,k~} is
undateral, temperate and polynomial of degree m on diagonais.
PaOOF. T h e converse follows immediately fl'om Proposition 2.7. If {ak,,~:~ } is unilateral
and polynomial of degree m on diagonals, then there is a unique bilateral array {bk,,~ }
which extends {ak,,k~} (i.e., bkl,k2 = ak~,k,2 , ~1, k2 ~ 0) and is polynomial of degree rr~ on
diagonals. Fhrthermore, if {ak,,k~ } is temperate, then {bk,,k2 } is temperate. By Proposition 2.7, there exists a unique DDO L satisfying
b~:~.~ = L(e ik'~ (c ~k2o)
kl, k2 ~ g .
Thus (2.11) holds. Since {ak~,k2 } has a unique extension to a bilateral array and Proposition 2.17 guarantees a unique DDO L, the DDO L satisfying (2.11) is unique. This
establishes Corollary 2.10.
We record for future reference the tbllowing formula which is an immediate consequence of L e m m a 2.5 and Proposition 2.7 (which asserts that (e-~k~ ~k~ ((p) (~r is a
polynomial in k).
Lemma
m
2.12. Let L = ).~ /3tD (~) be a DDO. ff ~ and ~ are in D, then
~=0
1
~ -ikO~
ikO\
k---* oc
The Schwarz nuclear theorem guarantees that there is a one-to-one correspondence
between the continuous linear maps L : 79 ~ ~p/ and the distributions u on the torus,
(0D) 2. This correspondence is implemented by
(2.13)
~((~(C i~1 )~p(e i~
= L(~) (~).
Proposition 2.7 and Corollary 2.10 give a characterization of DDO inside the space of
continuous linear maps L from 7) to 7) ~ in terms of the Fourier transform of the corresponding distributions u. (If (2.8) and (2.13) hold, then u~(-k~, k2) = ak~,k2). Another
characterization of D D O is provided by the following proposition.
P r o p o s i t i o n 2.14.
are equivalent.
The following conditions on a continuous linear mapping L: D ~ ~
(i) There exists an m _> 0 such that
( - X ) m - e ( me) e - i e ~ 1 7 6
g=0
(ii) L is a DDO.
(iii) The distribution u defined by (2.13) has support in the set {(e ~~ eie): 0 E R}.
PROOF. Observe that (i) implies that if the bilateral array {akl,k2 } is defined by
(2.15)
ak,,k~ =
L(e ~kl~ (~-~k2o) ,
then {a~ 1,k2 } is polynomial on diagonals. {akl,k2 } is temperate since L is continuous. By
Proposition 2.7, there exists a DDO L0 such that
(2.16)
akx,k2 = Lo(e ikl~ (e-ik2~ .
tf kl is fixed, then (2.15) and (2.16) imply that
Lo(eik~~
= L(eik:~
for all k2 E Z. Hence
(2.17)
Lo(e ikl~
= L(ciklO).
Since kl is arbitrary and L is continuous, (2.12) implies that L = L0. Thus, L is a DDO.
This shows that (i) implies (ii).
Suppose (ii) holds and u is defined by (2.13). If m is the order of L, then
U((1--ei(Ol-oa))mg~(ei~162
/7~
: ~ (--1)k (k)u(eik~176176162
k=O
m
= ~ (-1)k ( k ) L ( e ik~
(e-~k~
k=O
( d ~ m+l
=
\7-~)
L(10) (V~)
Thus, (iii) holds.
Suppose now that (iii) holds. Since (OlD)2 is compact, u has finite order, and thus,
there exists an rn _> 0 such that (1 - e i ( e l - ~
= O. For ~o, %bE/9,
~=0
g=O
77t,
~=0
= ~((1 - ~i(01-02/)m10~)
:0.
Thus (i) holds. This establishes Proposition 2.14.
~3. A m o d e l t h e o r y for m - i s o m e t r i e s
In this section, we shall model the general finitely cyclic m-isometry as multiplication
by z on the completion of a pre-Hilbert space of analytic functions with respect, to a norm
defined by a DDO.
Define 77a C 77 by
^
77~={~e77:10(~)=0 if ~<0}.
Since the Fourier transform theory implies that !o E 77,~ if and only if
~(~i~ =
oo
~ d(~0~ ~n~
n=0
with convergence in 77, it is clear that 77~ consists precisely of the boundary values of
analytic functions on D whose power series coefficients form a rapidly decreasing sequence.
We let P denote the canonical projection of 77 onto D~ defined by
(P10)~(") = { 10~('~)o ." ><o~
In like fashion, let D~a denote the space of analytic distributions defined by
~/a~-{UE~/:?t~(n):O
if
n<0}.
D~a can be r e g a r d e d as the space of b o u n d a r y values of analytic functions on D whose power
series coefficients form a t e m p e r a t e sequence. We let P denote the canonical p r o j e c t i o n of
79r onto D~ defined by
(pu)^(n)=
u (n)
0
n>_O
n<0
'
D e f i n i t i o n 3.1. A distribution Toeplitz operator (DTO) is a linear m a p p i n g A : Da --* :D~
t h a t has the form
A = PLID~
(3.2)
for some D D O L. If A is a DTO, we define the order of A to be the order of L where L is
as in (3.2).
Lemma
3.3. I f A and L are as in (3.2), then
A(~)(~) = L(p)(~)
whenever ~, ~ E 79a.
PROOF. We first show t h a t if u E 79r, then
Pu(r = u(~)
(3.4)
whenever ~ E Da. Accordingly, fix u E 79r and ~ E :Da.
k=0
=
=
Using (3.4), we now see t h a t
A(~)(~) = P(L(~))(~)
= L(~)(~)
which establishes L e m m a 3.3.
As a corollary to L e m m a 3.3 and Corollary 2.10, one sees t h a t if A is a DTO, t h e n the
DDO L satisfying (3.2) is unique. In particular, the notion of the order of A in Definition
3.1 is well-defined.
If L is a DDO, let us agree to say t h a t L is positive (L > 0) if
L(~)(~) > 0
whenever ~ E D. If A is a DTO, let us agree to say t h a t A is positive (A >_ 0) if
A(~)(~a) _>>0
whenever p E / ) a . We remark before continuing that if L is a D D O and A = PLIZ)~ is a
DTO, then it is possible t h a t A > 0 yet L is not positive. A n example is given by letting
L = D. This is a reflection of the operator theoretic fact t h a t unlike the case when m = 1,
not all m-isometries have extensions to invertible m-isometries.
L e m m a 3.5. Let L = k /~eD(~) be a D D O with [3m ~ O. If L > O, then/3m is a positive
~=o
measure. If A = P L I D~ > O, then/Jm is a positive measure.
PROOF. If L > 0, then A > 0. Thus, it suffices to show only the final assertion of the
lemma. Thus, assume A _> 0. Fix qo E 2P~. By L e m m a 2.12,
9m(l~l ~) =
lira
1 (e_ik oLeiko)(T)(~)
k~oc k - - ~
= lira
k--,oo
>_0.
1
A(e~kO~p)(eikO~)
Hence,
(3.6)
~m(l~l 2) ~ 0 whenever
~ E Do..
But,
{I~ol2:~ET)a}
isdensein
{~bED:~b>0}.
Hence (3.6) implies that ~m is a positive distribution and thus is a positive measure.
concludes the proof of Lemlna 3.5.
This
If L _> 0 is a DDO, we define the form space of L, H~, to be the completion of D with
respect to the sesquilinear form [., "]L on 7? • 79 given by
[~, ~] = L(v)(~).
Specifically, one sets A4L = {~a C 1): [~, ~]L = 0}, observes by the Cauchy-Schwarz
inequality that [., "]L induces a inner product on 7)/AJL, and lets H~ denote the completion
of ~l) /.A/[ L .
Likewise, if A > 0 is a DTO, we define the form space of A, H2a, to be the completion
of ida with respect to the sesquilinear form [., "]A on 7), x Z)~ given by
[~, r
= A(~)(~).
If L > 0 is a DDO, then one can attempt to define a linear operator ML on H~ via
the formula
(ML
)(e
=
In general, this operator need not be well-defined on D/AdL and, even if it is, may not
extend to a bounded operator on HL2.
Lemma
3.7. Let L be a D D O and fix c > 1. L > 0 and
ML
iS a well-defined bounded
operator on H~ of norm <<_c if and only if
(3.8)
0 <_ L - c - 2 e - i ~
PROOF. First assume that L > 0 and
IIMLll <
( L - c-2 e:i~ L ei~
e iO .
c. If ~ E I), then
= L(qo)(@) - c- 2L(eW p)(eWcp)
=
-
c
211ML
II
20
and thus (3.8) holds. Conversely, assume that L is a DDO satisfying (3.8). Note that by
conjugation each term in the following telescoping sum is positive:
k-1
L - c-2ke-ikOL
eikO = ~
c-23e-iJ~
- c - 2 e - W L e W ) e iJ~ .
j=0
Also note by Proposition 2.7 that
(e-ia~ e{k~
= (~L~)(eik~
-~k~
is a polynomial in k. Hence since c > 1
lim c-2ke-{k~
k~c~
eik~
= 0.
Consequently,
L ( ~ ) ( ~ ) = lim (L - c-2ke-ik~
eik~
~0
and we have established that L > 0. Calculating as before establishes that
>0.
whenever W E 79. This inequality establishes both that ML is well-defined on ~D/.~ L and
extends by continuity to an operator ML on H~ with
IIMLtl _< c.
This establishes L e m m a
3.7.
L e m m a 3.7 suggests that we introduce the following set for a DDO L r 0.
SL = {e > 0 : 0 <_ L - c--2e-Z~ e ~~
There are three possibilities which we shall consider in succession.
C a s e 1. S L : r
In this case, Lemma 3.7 guarantees that even though L might be positive
(so that HL2 is defined), either ML is ill defined on Z)/A.'IL or does not extend by continuity
to a b o u n d e d operator on H L"
2
C a s e 2. SL • r and inf SL > 1. In this case Lemma 3.7 guarantees that L _> 0 and ML
is a well-defined operator with
IIMLII = inf SL.
Note also that if ord (L) = 0, then (3.6) holds with c = 1 (i.e. 1 C SL). Thus o r d ( L ) _> 1
in this case.
C a s e 3. SL r r and inf SL _< 1. In this case necessarily if L _> 0, then ord (L) = 0. To
see this fix c <_ 1 with c E SL, and observe that if k > 0, then
k 1
L - c 2ke-ikOL eikO = E
c 2Je-iJO(L - c 2e ~~ ei~ ~J~ >_ O.
j=O
Hence if 9) E 7),
(3.9)
C(p)(@)
c 2k(e i~~176
Now if m > 1, then
lim
k~oc
1
~
L(qo)(~5)= 0
> (I.
Hence by (3.9) and Lemma 2.12, if L -- y]~ fleD(t) then
- Z ~ ( ~ ) ( 4 ) > 0.
But since L >_ 0, Lemma 3.5 implies that/3m > 0. Thus, ~m = 0 contradicting m _ 1.
We formalize the above observations into the following definition which precisely identifies those D D O L with the property that L >_ 0 and ML is a well-defined bounded
operator.
D e f i n i t i o n 3.10. A Dirichlet operator is a DDO L with the property that either there
exists a constant c > 1 such that
0 <_ L -
c-2e-i~
e i~
or ord (L) = 0 and L _> 0.
Note t h a t in Definition 3.10, the condition ord L = 0 and L > 0 is equivalent to the
assertion that L is a positive measure.
If c is a given constant, then the Dirichlet operators L with ]IMLH <_ c can be completely parameterized via an aifine transformation by the positive DDO as described in
the following proposition.
P r o p o s i t i o n 3.11. The Dirichlet operators of order 0 consist of the positive measures, f f
m >_ 1, the Dirichlet operators with ord (L) = m and I]ML]] <_ c consist precisely of the set
{ilk=0 c - 2 k e - i k e Q e i k ~
isa
DDO, o r d ( Q ) = m ,
Q>0}.
PROOF. The first assertion of the proposition follows from the definition of Dirichlet operators. To prove the second assertion, note that the definition implies that if L is a DDO
and o r d ( L ) > 1, then ]]ML]] > 1. Hence it suffices to assume c > 1. We first note that if
Q is a DDO and c > 1, then the series
E C 2ke-ikOQeiko
k=o
converges. This guarantees that the following calculations are justified.
First assume that L is a Dirichlet operator with ord (L) -- m and ]]ML]] <_ c. If Q is
defined by
(3.12)
Q=L-c
2c ieLe~~
then Q is a DDO, ord Q = m and Q > 0. Conjugating (3.12) by c - k e ik~ and s u m m i n g in
k yields t h a t
3--1
~-~ c - 2 k e - ~ k ~
*k~
L - c-2Je-*J~
~j~
m
k=0
Since L is a DDO and c > 1, Proposition 2.7 implies t h a t
lim c - 2 J e - i J ~
ijO = O.
Hence,
L = ~
c-2ke-ik~
ik~ .
k=0
Conversely, assume t h a t Q is a DDO, ord (Q) = m and Q _> 0. Set
L = ~
c-2ke-ikOQe
ikO .
k=0
Computing, we find t h a t
L-
c-2e-i~
i~ = Q > 0
and L is a Dirichlet o p e r a t o r with IIML tl <~ c. This establishes Proposition 3.11.
If L is a Dirichlet operator, then ML is an operator on H 2 and thus ( y x - 1)e(ML),
as defined in Section 1, is an operator on HL2 . The following proposition gives a formula
for the sesquilinear form induced by this operator.
Proposition
3.13. I l L is a Dirichlet operator, g >> O, and qo, r E 79, then
[(yx - 1 / ( M c ) ~ ,
~]L =
L
(~)(~).
PROOF.
k=O
k=0
=
~
L
(~)(~})
(Lemma 2.4 i t e r a t e d e t i m e s ) .
This establishes Proposition 3.13.
We finally are able to state and prove the first model theorem of this section.
Theorem
3.14. /f Tt is a Hilbert space, m ~ 1, T E s
is an invertible m-isometry,
and 2" E ~f is a bicyclic vector for T, then there exists a unique Dirichlet operator L with
ord (L) < m - 1 and a Hilbert space isomorphism U: ~ --~ H~ such that
T = U*ML U
and
U(~/) : 1.
Conversely, if L is a Diriehlet operator with ord(L) = m -
1, then ML is an invertible
m - i s o m e t r y with bicydie vector 1.
PROOF. First note that if L is a Dirichlet operator, then by the definition of H 2, 1 is a
bicyclic vector for ML (convergence in 19 implies convergence in H~). Also note that if
ord (L) = m - 1, then
a fact which implies via Proposition 3.13 that ML is an m-isometry. Since I9 C_ ran ML,
ran ML is dense and hence M c is invertible by L e m m a 1.21.
Conversely, let 7-/, T, and 2' be as in the statement of the theorem.
m-isometry, the infinite bilateral array {ah,k 2 } defined by
Since T is an
akl,k2 = ( T h ' ) ', Tk2~}
is t e m p e r a t e and polynomial on diagonals of degree < m - 1. Hence, by Proposition 2.7,
there exists a D D O L with ord (L) < m - 1 such that
(3.15)
(Tkl"7, Tk27> : L (e ih~
(e -{k2~
for all integers kl, k2.
By taking linear combinations of (3.15) and using the continuity of L and Proposition
1.20 we deduce that
(3.16)
<~(T)-y, r
: L(~)(~)
for all ~, r E i9.
We first show that L is a Dirichlet operator. If ord (L) = 0, then (3.16) implies L > 0
and hence by the definition L is a Dirichlet operator. If ord (L) > 1 observe t h a t (3.16)
implies that if ~ E 19, then
(L - c-2e-i6Le{~
: <(i - c ~T*T) ~(T)7, r
_>0
whenever c _> IITIt (note: since ord(L) > 1, T is not an isometry; thus tlTII > 1), and
again L is seen to be a Dirichlet operator.
Now, since L is a Diriehlet operator, H~ and ML are well-defined by the remark
preceding Definition 3.10. To construct U: ~ -~ H~, densely define an operator U0 on
by the formula
(3.17)
Uo(~(T)7) = ~ C H ~ .
Observe that (3.16) implies that Uo is well-defined and extends by continuity to a Hilbert
space isomorphism U: % / ~ H~ (U0 is in fact defined densely, since ~/is bicyclic for T and
U is surjective since " D / M L is dense in H~).
Finally, note that by definition U(~) = 1 and that (3.17) implies U T - M L U . This
concludes the proof of Theorem 3.14.
If 7~ is a Hilbert space, T C s
is an invertible m-isometry and ~ C 7~ is a nonzero
vector, then 3' is bi-cyclic for T I V{Tk7 : k C Z} and so there exists a uniquely associated
Dirichlet operator given by Theorem 3.14. We will denote this associated Diriehlet operator
by (T, 7)^.
The remainder of this section is devoted to three tasks. Firstly, we shall list without
proof the rather straightforward analogs of Lemma 3.7, Definition 3.10, Proposition 3.11
and Proposition 3.13 for DTO, and Theorem 3.14 for the case of rn-isometries not assumed
to be invertible but possessing a cyclic vector. We then continue on to formulate notation
that allows one to model multi-cyclic and multi-bicyclic m-isometrics. This involves little
more than allowing the coefficients in DDO and D T O to be matrices and the form spaces
associated with them to be completions of I) ~ and 79~:, respectively. Finally. we close the
section with a change of cyclic vector formula which turns out to have great value in the
application of the model theory to the operator theory of m-isometrics.
L e m m a 3.18. Let A be a D T O and fix c > 1. A >_ 0 and MA is a well-defined bounded
operator on H 2 of norm < c if and only it"
(3.19)
0 < A - c 2e-i~176
The following definition precisely identifies those DTO A with the prot)erty that A > 0
and MA is a well-defined bounded operator.
D e f i n i t i o n 3.20. An analytic Dirichlet operator is a DTO A with the property that either
there exists a constant c > 1 such that
O<~ A - c
or ord (A) = 0 and A >_ 0.
2e i~176
P r o p o s i t i o n 3.21. The analytic Dirichlet operators of order 0 consist of P L ]Da where
L is a positive measure. I f m > 1, the analytic Dirichlet operators with oral(A) = m and
liMa l[ <- c consist precisely of the set
Proposition
3.22. I r A is an analytic Dirichlet operator, g > O, and ~o, ~ E Da, then
g
T h e o r e m 3.23. / f 7-/ is a Hilbert space, m > 1, T E s
is a m-isometry and 7 E 7-I
is a cyclic vector for T, then there exists a unique analytic Dirichlet operator A with
ord (A) < m - 1 a n d a Hilbert space isomorphism U : 7-f -+ H~ such that
T = U*MAU
and
=
1.
Conversely, if A is an analytic Dirichlet operator with ord(A) = m - 1, then MA is an
m - i s o m e t r y with cyclic vector 1.
If T E s
is an rn-isometry and ~ E 7-/one can set
u , = V { T % : k _> 0}
and
W i t h this setup, Ty is a cyclic m - i s o m e t r y and 3' is a cyclic vector for T-r Accordingly by
Theorem
3.23, there exists a unique analytic Dirichlet o p e r a t o r A with the p r o p e r t y t h a t
(T~, 7) is u n i t a r i l y equivalent to (MA, 1). We shall let (T, 7) ~ denote the unique D D O such
t h a t A = P ( T , ~/)^ [ Da.
There is a certain amount of ambiguity in the (T, 7)~ notation. Observe t h a t if T E
s
is an invertible m - i s o m e t r y and 7 E 7-I then according to the r e m a r k following the
proof of T h e o r e m 3.14 (T, 7) ^ is the unique Dirichlet o p e r a t o r such t h a t
( ~ ( T ) % ~ ( T ) 7 ) = (T, 5 ) ' ( ~ ) ( ~ )
for all ~a, ~p c D. On the other hand, (T, 3')~ as just defined in previous paragraph has the
defining property
(~(T)',/, r
= A(~)(~)
:
(T, ~)'(~)@)
for all p, ~) E D~. Recalling the remark following the proof of L e m m a 3.3, we find that
these two definitions of (T, ^f)^ agree. The operator theory content of this observation is
recorded without proof in the following proposition. If T E s
is invertible and 224 is a
subspace of 7-/, let us agree to say that AA is bicyclic for T if the smallest subspace of 7-/
containing 2t4 and invariant for T and T - 1 is 7-f.
and T2 E s
be invertible m-isometrics. Let
P r o p o s i t i o n 3.24. Let T1 E s
Adz C 7-/1 and Ad2 C_ TI2 be bicyclic invariant subspaces for T1 and T2 respectively. If
U : Ada --* A42 is a Hilbert space isomorphism such that UT1 = T2U, then there exists a
Hilbert space isomorphism O: ~1 -~ 7-t2 such that (IT1 = T20 and U = (I 1A41.
Before continuing we explore some of the implications of Theorem 3.23 for cyclic
2-isometrics. Our first observation is that Proposition 1.23 implies that there are no nonunitary invertible 2-isometries. This implies via Theorem 3.14 that there are no Dirichlet
operators of order 1. Accordingly, the models for 2-isometries are given only by analytic
Dirichlet operators.
Now, if T C E(7-t) is a 2-isometry and ~, E 7-(, then since the unilateral array
(Tk]~/, Tk23`) is linear on diagonals it is natural to define the slope p and intercept
of (T, 3`) to be the elements of D' defined by the formulas
(3.z5)
[ (Tk+13`,T3`) - (Tk%3`)
/~ (k) = ~ (T~/,T-k+17}
(~/,T-kT)
k > 0
k < 0
and
(3.26)
k3`,~)
//(k) =
{(T
<%T-k@
k>0
k < 0
On the other hand there exist a pair of distributions r
A = P(~ID +
and [Jl such that;
9o) I ~ ,
Furthermore, the formulas that relate the above distributions are given by
~1 =
~ :
and
(3.27)
~o = / ~ +
(1 -
P)(D#).
Now if one i8 given a 2-isometry T E s
then by varying the choice of vector 7
one can o b t a i n various slopes and intercepts. Three choices have been considered. In [R2]
Richter chooses V c ker T*. Evidently,via (3.26) one sees t h a t this corresponds to requiring
the intercept of (T, y) to be a scalar multiple of d . In Section 6 we shall analyze the case
where y is chosen so t h a t the slope of (T, 7) is d .
In Section 8 we shall investigate the
case where the slope and intercept are chosen colinear.
Before continuing we record the following simple reformulation of L e m m a 3.18 in the
case where ord (A) = 1.
Lemma
3.28. Let A = P(/~ID + ~o) I1)a be a DTO. A is an analytic Dirichlet operator
if and only if there exists c > 0 such that
(3.29)
C P ~ l l Z)~ ~ A .
D1rthermore, if the covariance of A (coy(A)) is defined by
coy(A) = llAMA ]1�89 ,
then
We now t u r n to the problem of modeling multicyclic m-isometries. We shall do the
case of invertible m-isometries in some detail and merely state the n o t a t i o n and results for
the noninvertible case.
If nl a n d n2 are positive integers, recall t h a t T~"1,n2 denotes the space of n l x n2
matrices with entries in 79. Likewise, let D D O ~ l'n~ denote the space of nl • n2 matrices
with D D O entries L~s with the order of L~s _< m.
~* =
(~1,
.- ,~,~).
Our first observation is t h a t (Dl'n) ' can be identified with (D') n,1 via the pairing
(3.30)
u(r
= ~22 u,.(r
where r = (r
G l ) 1''~ and u = (u,,) e (D') n'l. Note also, t h a t if L = ( L ~ ) e D D O ~ n
then L induces a linear m a p p i n g from l ) ~'1 into (T~) n'l via the formula
Combining (3.30) and (3.31) leads to the formula
(3.32)
= E
L(~)(r
L~s(p~)(~)
r8
for all ~v, r E/)~,i.
We now p o i n t out that with the notations of the preceding paragraph that all of
the results about scalar DDO of Section 2 have analogs for the case of matrix DDO. In
particular, note that if L E D D O ~ ~, then there exist unique/3t E (~D')n'~ with
L = ~
~eD (t)
~=0
and if 3~D L is defined as in Definition 2.3 by setting
d
L = E
dD
g/3eD(~-l)
then the following three formulas hold for all ~, ~ E ~n,1.
e-i~
(3.33)
'~
L =
d L.
dD
(d) m
(3.34)
(3.35)
5"~ { L ( e i k ~ 1 6 2
lim
Ik:o =
~
L(~)(r
1
Furthermore, the formula
(akl,k2~t,v} = L(eiklO~t) (e ik2Ov*) , u, 11 C
C n'l
sets up a 1 - 1 correspondence between n x n matrix DDO L of order m and bilateral
(resp.. unilateral) temperate n x n block matrix arrays which are polynomial on diagonals
of degree rn as in Proposition 2.7 (resp. Cor. 2.10).
Finally we record the following generalization of Proposition 2.14. Note that if L :
D n,1 ~ (:DI) n,1 is a continuous linear mapping, then a matrix distribution can be defined
via the Schwarz nuclear theorem by requiring
(3.36)
u(~(ei~162176 *) = L(~o)(r
~, r C
,~n,1
.
Proposition 3.37. The following conditions on a continuous linear mapping L: D n,I -~
(~y)n,1 are equivalent.
(i) There exists an m > 0 such that
~-~ ( - 1 ) m e ( f , ) e
~e~176
e=O
(ii) L is a DDO.
(iii) The distribution u defined by (3.36) has support in the set
{(~)o, e~o): o E ~ }
If L E D D O ~ n, let us agree to say that L is positive (L > O) if L(~)(~*) >_ 0 whenever
E ~Dn'l �9
If L C D D O ~ n, we define the form space of L, H 2, to be the completion of ~ , 1 with
respect to the sesquilinear form [., "]L on 7)~'1 x /3 ~'1 given by
[~,~]L = L(~)(~*)
The following lemma describes when multiplication by e i~ on H 2, ML, is well-defined
and bounded. Its proof is identical to that of Lemma 3.7.
L e m m a 3.38. Let L E D D O ~ n and fix c > 1. L > 0 and M L is a well-defined bounded
operator on H2L of norm < c if and only if
0 < L - c-2e-i~
(3.39)
~~ .
As in the case of scalar DDO, we say that L C DDO~d *~ is a Dirichlet operator if either
there exists a constant c > 1 such that (3.39) holds or ord (L) = 0 and L >_ 0. It is then
the case that L E DDO~d '~ is a Dirichlet operator if and only if L >_ 0 and ML is bounded.
If c > 1 is a given constant, then the Dirichlet operators L E D D O ~ n with HMLH
c can be completely parameterized via an affine transformation by positive D D O ~ ~ as
described in the following proposition. The proof is identical to that of Proposition 3.11.
Proposition 3.40. I f m >_ 1, the Dirichlet operators L
DDOn~ n with [[ML][ <_ C consist
E
precisely of the set
{~-~ c - 2 k e - i k ~ 1 7 6
V
~,n
DO,~
~
Q >_ O }
.
k=O
If L E DDO~d ~ is a Dirichlet operator, then ML is an operator on H 2 and thus
(yx - 1)e(ML), as defined in Section 1, is an operator on H~. The following proposition
gives a formula for the sesquilinear form induced by this operator. It follows from (3.33)
in exactly the same way that Proposition 3.13 followed from Lemma 2.4.
Proposition
3.41. I l L E DDO~A n is a Dirichlet operator, f >_ 0 and ~, ~ E ~)n,1, then
[ ( y x - 1)e(ML)~,~]L =
~-~
C (~o)(r
For 1 < r < n, we define er E I) n'l to be the vector whose r t h
coordinate is 1 E ~D
and whose other coordinates are O.
We finally are able to state and prove the model theorem for finitely bicyclic invertible
m-isometries. For T E s
us agree to define p(T) E s
7 = (7~) E ~t.~n,1, and p a 1 x n row vector of polynomials let
1'~ by
p(T)
Since s
:
(pr(T)).
Ln acts n a t u r a l l y on 7f n'l this leads to the notation,
p(T)7 = E
(3.42)
p<(T)"/<.
In p a r t i c u l a r if T is an invertible m-isometry, then by Proposition 1.20, T possesses a
/)-functional calculus and by continuity (3.42) becomes
9~(T)7 = Z
(3.43)
Theorem
~(T)7~
,
~ E/9"~.
3.44. f f ~ is a Hilbert space, m > 1, T E s
"Y = (~/r) E ~ , 1
is an invertible rn-isornetry, and
is an n-bicyclic tuple for T, then there exist a unique Dirichlet operator
L E D D O ~ _ I and a Hilbert space isomorphism U : ~ --~ H2L such that
T
= U*MLU
and
U('y,.) = e,.
1 < r < n.
Conversely, if L E DDO~d~ 1 is a Dirichlet operator, then ML is an invertible m-isometry
with bieyelic n-tuple (e~).
PROOF. F i r s t note t h a t if L E DDO~*~ 1 is a Dirichlet operator, then by the definition
of H I , e l , . . . , en are bicyclic vectors for ML (convergence in ID1'~ implies convergence in
2 Also note t h a t
HE).
d
L=O,
a fact which implies via Proposition 3.41 that ML is an m-isometry. Since 19n'l c_ r a n ML,
ran
ML is dense and hence ML is invertible by Lemma 1.21.
Conversely, let 7-/,T, and 71,... ,Tn be as in the statement of the theorem. We
construct L E DDOm_ 1 componentwise. Accordingly, fix r, s ~ { 1 , . . . , n}. Since T is an
m-isometry, the infinite bilateral array {a h,k2 } defined by
a~:~,k~ = (Tk~7,-, Tk2%}
is temperate and polynomial on diagonals of degree < m - 1. Hence by Proposition 2.7,
there exists a DDO LTs with ord (L) < m - 1 such that
L,.~(e{h~
(3.45)
-{k2~
= (Tk'%,
Tk2"/~-)
for all integers kl, k2.
By taking linear combinations of (3.45) and using the continuity of L and L e m m a 1.20
we deduce that
(3.46)
Lrs(T)(r
= (~(T)%, r
for all ~, r E "D.
Set L = (Lrs) E DDOm_ 1. If ~, r 6
L(~)(r
= ~
then
L~s(~s)(~}T)
r,s=l
: E
(~s(T)%, r
r~8
= (~o'(T)7, r
a fact which we record as
(3.47)
L(~o)(r
= (~(T)7,
r
If ord (L) = 0, then (3.47) implies L > 0 and hence by definition L is a Dirichlet
operator. To see that L is a Dirichlet operator when ord (L) > 1 observe that (3.47)
implies that if ~o C 79n'l and c = [ITH, then
(L - c- 2e-i~ Le~~
*) = ((1 - c- 2T*T)~(T)'-/, ~T (T)',/)
>_0
(note: since ord (L) _> 1, T is not an isometry; thus
IITII
> 1). Thus, L is a Dirichlet
o p e r a t o r and b o t h H~ and ML are well-defined.
To construct U: ~ ~ H~, densely define an operator U0 on H by the formula
(3.48)
Uo
~,.(T)%.
= (~,.) e H~.
r=l
Observe t h a t (3.47) implies t h a t Uo is well-defined and extends by continuity to a Hilbert
space isomorphism U : 7-/-~ HL2 . Uo is in fact defined densely, since 3`1,. � 9 , % are bicyclic
for T and U is surjective since 79n'l / {9~ E/)n,1 : L ( V ) ( ~ , ) = 0} is dense in H~.
Finally note t h a t by definition U(3`j) = ej and t h a t (3.48) implies U T = MLU. This
concludes the proof of Theorem 3.44.
If 7-/ is a Hilbert space, T E Z;(~) is an invertible rn-isometry and ~/1. . . .
, 3`n C ' ~ ,
then 3' = (3`1,-.. , 3`~) is a bicyclic n-tuple for
TI V{Tk3`j: k E ~, l_<j_<n}
and so there exists a uniquely associated matricial Dirichlet o p e r a t o r given by T h e o r e m
3.44. We will denote this associated matricial Dirichlet operator by (T, 7) ~.
Letting D T O TM''~2 denote the space of nl x n2 matrices with D T O entries A,.~ with
the order of A~s < m, we can then define
A(~)('~*)
positivity of
A, the form space of
for
~, r E D2 '~ ,
A, H~, MA,
multiplication by
c i~
on
H ~ and
matricial analytic Dirichlet operator in the obvious way. The straightforward analogs of
L e m m a 3.38, Proposition 3.40, Proposition 3.41 hold and lead to the following model
theorem for multicyclic m-isometrics not assumed to be invertible.
Theorem
3.49. If 7-( is a Hilbert space, m _> 1, T E s
is an m-isometry and ~/ =
(3`r') E ~_~n,1, is a n-cyclic tuple for T, then there exists a unique analytic Dirichlet operator
A E D T O ~ _ I and a Hilbert space isomorphism U: ~ ~ H~ such that
T = U*MAU
and
U(%.) = e r
1< r <n.
Conversely, if A E DTO~2'~I is an analytic Dirichlet operator, then MA is an m-isometry
with cyclic n-tuple (e~).
Now, as in the case of invertible rn-isometries, we introduce the following notation. If
T E g ( ~ ) is an m-isometry, 3` = (3`~,... , 3`,~) E ~n,~ is a choice of n vectors and ~-~ c
is defined by
~
= V { r k 3 ` ~ : k >_ 0 ,
l<r<n},
then T [ H~ is an m-isometry and ~, is a cyclic n-tuple for T 17-/~. Accordingly, by Theorem
3.49 there exists a unique A c D T O ~ _ I with the stated properties of that theorem. In
the sequel we shall define (T, 1')^ to be the unique DDO such that A = P(T, "7)^ 117an,1.
Now, suppose one is presented with a finitely cyclic m-isometry T and wishes to use
Theorem 3.49 to model T. Since in general T will have many cyclic n-tuples, there are
many associated D T O A. This phenomenon is already present in the case of isometries.
To be precise, let T be a cyclic isometry. Thus, T is unitarily equivalent to M~,o on H2(#)
for some positive measure # supported on the circle. In this case a well-known piece of
function theory gives that
H2(#) = L2(#~) | H 2 w
where d# = d#~ + w dO is the Lebesgue decomposition of #. Furthermore,
H2 w ~
1 H2
if
L2 w ~
if
/
lnw
dO
>-oc
Inw ~=-oc
where f c H 2 is the outer function such that Ifl 2 = w. Assume that T is not unitary so
dO
that f In w 7~
> - c o . With this model, a vector I' = P @ g is cyclic for T if and only if
fg is outer and qo 7~ 0 for #~ a.e. e i~ Furthermore, it is easy to check that
(3.50)
dO
(T'1')^ = 1~12d#~ + Igfl2 2--~"
Of course, in the theory of isometries, one knows in advance via the wandering subspace
argument that 1' can be chosen so that g f = 1 in (3.50), so that the pure cyclic isometry
is unique and unitarily equivalent to Memo on H 2. When m k 1, however, there is no
preordained choice for 1' (recall the remarks made in the paragraph following (3.27)) and
analogs of formula (3.50) have added interest. The following theorem shows how (T, 1')^
transforms when 1' is transformed via the functional calculus.
For F = (FTs) C "~n2,nl define F " and F* by
F " = (F~r) ,
and
F * = (P~r).
Now if F E T~ n l then F induces a map F : :Dn l J -~ l;)n2,1 so that if L E D D O ~ 'n2 then
L F : :D'~'1 -* (:Dr) ~3,1. Concretely,
LF
=
LrtFts
�9
/
Similarly, if G E :Dn4'na , then G induces a map G: 7?'~'~'1 ~
~DIn4'l so that GL represents
a map GL: :D~2'1 -* (Z)') n~'~.
Now recall that if T is an m-isometry, 7 E H n t , 1 and ~ E 7)1,*~
--a , then p ( T ) 7 E H is
defined by
~(T)~ : Z
~(T)%~.
8
More generally, if F E T~ ~.... define F ( T ) 7 E ~/n2,1 by
Theorem
3.52. Fix positive integers m, rq, and n2. If T is an m-isometry, 3 E ~,~,1,
F E 7?~2''u and g >_ O, then
(3.53)
~
(T,F(T)~/) ^ = F ~*
(T, 7) ^ F ' .
PROOF. Let # = F ( T ) ~ and set L = (T,'~) . We first show that (3.53) holds when f = 0.
Thus we need to show that
(T, #)^ke = (Fr*LF~)~:t
for all k, ~ _< n2. But if ~, ~p E 79~, then
(T, #)^ke(~0)(r = {~(T)#t, r
= (~(T)E~ F~(Tb~,r
(3.47)
Fks(T)~l
(3.51)
= Er,s (((pFt~)(T)"/T, (r
= E~,~ L~,.(~Fe,O(r
(3.47)
= (F~*LF)k~((p)(~)
(3.51).
To deduce t h a t (3.53) holds when g > 0 simply apply the identity
d G*LG=G*( d
d--~
~-~ L ) c
which holds for all L E DDO~n~'~ and G E ~ 1 , n 2 . This establishes Theorem 3.52.
~4. Smoothing applied to m-isometrics
In this section, we describe how to convolve smooth functions with DDO, D T O and
m-isometries T. This allows one to reduce many qualitative questions concerning the operator theory of m-isometries to the case where the m d s o m e t r y possesses a model based
on a D D O with C ~ coefficients, i.e. a classical differential operator. A concrete example
of this p r o c e d u r e will be given in Section 7 where we shall show using the Arveson Extension T h e o r e m t h a t the model theory for 2-isometries can be u n d e r s t o o d in terms of a
Disconjugacy T h e o r e m for the underlying DTO.
Recall t h a t if ~ E l ) and t E R, then the translation of ~ by t, ~t, is defined by
r
i0) : 99(ei(O-t)) for 0 E 1~. If u E Z)', then the t r a n s l a t i o n of u by t E IR, ut, is the
d i s t r i b u t i o n defined by u t ( ~ ) = u ( ~ - t ) . Finally, ifw E I9 and u E I)', then the convolution
of w and u, w * u, is defined by the formula,
(~* ~)
:
[ ~(e'~) ~ 2 ~
These notions can be extended to translations and convolutions of elements of l ) T M ' n ~ and
(I9') n~,~2 componentwise.
We now extend these notions to DDO.
D e f i n i t i o n 4.1. If L : l ) ~'1 --* ( l ) 0 ~'1 is continuous and linear and t E R, t h e n we define
the t r a n s l a t i o n of L by t, Lt: Z) n'l ~ (ZY) n'l, by the formula
Lt(F) : ( L ( ~ - t ) ) t .
Proposition
4.2. Fix t E • and integers m > 0, n > 1. I f
L : ~ 3 t D (~1 E D D O ~ ~,
t=O
then
(4.3)
Lt = ~ (fi~)tD (t) �9
~:0
In addition, i l L is a Dirichlet operator, then Lt is a Dirichlet operator with []MLt ]I : IIMLII
and if A : P L [ I)a is an analytic Dirichlet operator, then At : P L t [ 19~ is an analytic
Dirichlet operator with lIMA, I1 = ]tMA]t .
PROOF. F i r s t recall t h a t D commutes with translation. Computing, we find t h a t if 9~ E
~D~'1 and r E l ) l'n
(/~D(~))t(99)(/~)-- (3~D(s
t))t(1~)
= 3eD(e)(~_t)(r
= (3e)t((D(e)~)~)
: ((ge)~D(~))(~)(r
Thus (4.3) holds� 9
Suppose now that L is a Dirichlet operator. If ord (L) = 0, then L > 0. Consequently
ord (Lt) - - 0 and Lt > O. Hence Lt i s a Dirichlet operator. On the other hand iford (L) > 1,
then by (3,39), there is a c > 1 such that
(4,4)
0 <_ L - c-2e-~~ Le ~~ .
If ~ E D n'l, then
( Lt - c - 2 e - i ~ Lt e~~
*) = L ( ~ - t ) ( p * t ) - c - 2 Lt(ei~ ~)( (e~~P) * )
-- L(~_t)(~**) - ~-~L(~(~247176247
*)
= L(~-t)(~*-t) - c 2(e-i~176
>0.
Thus, Lt is a Dirichlet operator. Furthermore since tlML]] is the smallest c such that (4.4)
holds, this computation shows that in fact ]]ML]] = ]]ML~ ]].
The proof of Proposition 4.2 is now completed by doing an analogous argument with
L replaced by P L ]~D,~.
Now, Proposition 4.2 implies that if T E s
is an m-isometry, ? E 7i ~A and t E R,
then ((T, 7 ) ^ ) t i s a n • n matricial Dirichlet operator and thus via Theorem 3.44 models
some operator Tt.
Proposition
4.5. L e t T E ~ ( ~ ) be an m-isometry, 7 E ~ , I
and t E R. I f Tt = e U T ;
(T~,~)~ = ((T, ~)~)~.
PROOF.
First note that ~-t(T) = ~(Tt). Thus, for ~, y~ E q')n~l
~ a , w e find that
= (~L(T)v,
,,~,(T)~)
= (~(Td%
'/'T(T~)V)
This establishes Proposition 4.5.
D e f i n i t i o n 4.6 If 0~ E 2) and L E D D O ~ n, then the convolution of w and L, ~ * L :
D ~,1 -~ (T)~)~'1 is given by the formula
.
(4.7)
w *L =
�9
w(cit)Lt dt
27c
If A is a D T O , t h e n co, A: ~n,1 __+ (T)Ia)n,1 is given by
(4.8)
co * A =
co(eit)At ~-~.
Note t h a t if A = P L 17)~ ,1 a n d w C D t h e n a n easy consequence of Definition 4.6 is
n,n , ~ C ~)n,1 a n d
t h a t w �9 A = P ( w * L) I :Da. Also observe t h a t if L = ~ fleD (e) C D D O m
e
r E 791''~, t h e n
(w �9 L ) ( ~ ) ( ~ ) =
(4.9)
Proposition
w(eit)Lt(~)(r
~-~.
4 . 1 0 . /if L = ~ fleD (e) E D D O ~ n a n d w E Z), t h e n co, L = ~ ( c o �9 fl~)D (e).
PROOF. If ~, r C ~)n,1, t h e n
(co,
= f
dt
(4.7)
dt
f
(4.3)
dt
: (E
T h i s c o m p u t a t i o n establishes P r o p o s i t i o n 4.10.
Proposition
4 . 1 1 . Fix w E 7) with w >_ O. If L is a Dirichlet operator, then w * L is a
Dirichlet operator and IIM,~,LN <_ IIML[I. I r A is a n analytic Dirichlet operator, then w * A
is an analytic Dirichlet operator a n d IIM~,AII <_ IIMAII.
PROOF. If ord L = 0, t h e n L _> 0. Consequently, w * L >_ 0 a n d ord (w * L) = 0. T h u s
03 �9 L is a Dirichlet o p e r a t o r if ord L = 0.
Now s u p p o s e ord L > 0. If c > 1 a n d
L - c-2e-i~
ie > 0,
then
w* L-c
2e io(w. L)eiO= f wLt -dt~ - c. 2e iO I w L , - ~dte ~o
: .f
dA
>0
by Proposition 4.2. Hence w * L is a Dirichlet operator and HM~,LII <_ HMLH. The proof
of the second assertion of the proposition follows similarly.
Now if T 6 s
is an m-isometry and 7 6 7-P~'1, then L = (T, 7)" E DDO,~n_I and
A = P L [ D ~ '1 is an analytic Dirichlet operator. Hence by Proposition 4.11, if w > 0, then
M~,L is a m-isometry. To identify M~,L spatially in terms of T and w we introduce the
following notations. If # is a positive measure and %/ is a Hilbert space, we let L ~ (#)
denote the Hilbert space of 7-l-valued p-measurable functions F such that
f ttV(x)ll~du(x) < ~ .
If # is supported on the circle and T e s
define T u e s ( L 2 ( # ) ) by setting
(Tt, F ) (e 't) = e~tT(F(ei') ) .
In addition, if ~ = (~'r) C ~n,a define Z. ~ (L.~(U))n'~ by requiting
("@.)r(e ~t) = "7,- for each
r _< n.
We now are able to identify Mw,L.
P r o p o s i t i o n 4.12. Let T r s
Let #
=
[d
be an m-isometry, ~ r ~ , 1
z~" T~ is an m-isometry and (T~, % )
dt
^
and ~ ~ T) with w > O.
= w �9 (T, ~) .
^
PROOF. If ~o, ga r D n,1
a , then
(~ �9 (T, ~)~)(v)(r
: / ( ( T , ~)~)~(~)(r
d~
: f ( T , "Y)^(qo-t)(r
d#
: / < ~_~(T)~, r
=/<
> d~
~(e"T)% ~(e"T)~ > d~
= (v~(T,.)'r~,, r
This computation establishes Proposition 4.12.
Now, by a m a t r i x measure p we shall mean an n x n m a t r i x ( # ~ ) with entries in the
complex Borel measures s u p p o r t e d on the unit circle and such t h a t
_> o
for all Borel sets A. If p is a m a t r i x measure and h E C n, then define a scalar measure
#h,h by setting
#h,h(A) =
( p ( A ) h , h}.
Thus, ~th,h is a positive measure for each h E C ~. Let us agree to say t h a t a m a t r i x
measure # is weakly definite if #(0D) > 0 (i.e. p(0D) is a strictly positive definite matrix).
Observe t h a t trivially if # is a m a t r i x measure, # is weakly definite if and only if #h,h r 0
for all nonzero h E C n. Finally, we r e m a r k t h a t L e m m a 3.5 has the following i m m e d i a t e
generalization.
Lemma
4.13. If L = s
'~'~ and L > O., then f,,~ is a m a t r i x measure. I f
feD(e) E D D O m
f=0
A = P L I 7 9 n,1 > O, then f,~ is a m a t r i x measure.
D e f i n i t i o n 4.14. If L = ~ f e D (e) E DDO n'n, then L is smooth if each fie is a m a t r i x of
s m o o t h functions. L is regular if L is s m o o t h and f m > 0 on 0D (i.e., the n x n m a t r i x
f m ( e i~ is strictly positive definite for all e i~ E cgD where m = o r d (L).
Proposition
4.15. L e t L = s
feD(~) E DDO~s n and let co E l). w * L is smooth. I f
f=0
L > 0 and co > 0, then co. L > O. I l L > O, f m is weakly definite, and w > O, then co* L
is regular.
PROOF. Let L = ~ fleD (e). Since
w*L=E
( w * f i t ) D (~)
and since w �9 fe, the convolution of a smooth function and a distribution, is smooth, co �9L
is smooth. If L > 0 and co > 0, then the positivity of co �9 L follows from (4.7).
Now assume t h a t L > 0, f m is weakly definite, and w > 0. By P r o p o s i t i o n 4.10, we
need to show t h a t w* f m > 0. Using the definition of convolution, and the facts t h a t w > 0
and f m is weakly definite we see t h a t
((co �9
h) = f
> O.
I
Hence (w * f l m ) ( d t) > 0 and the proof of Proposition 4.15 is complete.
We r e m a r k before continuing that Proposition 4.10 remains true if the D D O L is
replaced by a D T O A.
As we have already discussed if one is given an n-cyclic m - i s o m e t r y T then there are
in general m a n y choices of n-cyclic tuple and thus many possible models for T of t h e form
ML. Let us agree to say t h a t an n-cyclic m-isometry T E s
possesses a Wiener-Hopf
form if there exists an 7 E 7t ~'1 such t h a t
m
-
2
dO D(m_l) + E
(T, 7) ^ = 2--~
(4.16)
'BeD(e) ' 6=0
and
(4.17)
y
is an n-cyclic tuple for
T.
Theorem 4.18. ff T E L;(~) is an n-cyclic m-isometry and is there exists an 'n-cyclic
^
tuple 7 for T such (T, 7) is regular, then T possesses a Wiener-Hopf form.
PROOF.
Let
rn-1
(r, 7; = Z
"6D(~)
6=0
with a6 E 7?~ " and c~,~ 1 > 0.
By Wiener-Hopf factorization, there exists an outer
G E 7?2 'n such t h a t
c(eio)'<c(eiO
) = (~m_l(~iO)T) -1
Let # = G ( T ) 7 ((3.51)). Since O -1 C IDa , Proposition 1.20 and the cyclicity of 7 implies
t h a t # is a cyclic n-tuple for T. Set
rn--1
(T, .)^ = ~
~6D(6)
~--0
To see t h a t ~,~ 1 ~ - 1, note t h a t
1
i
(m- i)!
~)
(m,~)
,,,7~)
(T,,~) c ~
This establishes Theorem 4.18.
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