412
SPECTRAL SETS A N D N O R M A L
D I L A T I O N S OF O P E R A T O R S !
By B É L A S Z . - N A G Y
1. Spectral sets
In all that follows 'operator' will mean a linear operator on Hilbert
space, everywhere defined and bounded. The notion of spectral set is
due to von Neumannt5] : a set 8 of points in the plane of complex numbers
is a spectral set of the operator T if, for any rational function r(z) which
is bounded on 8, the operator r(T) exists and the inequality
\\r(T)\\< snp|r(z)|
(1)
holds; the condition of the existence of r(T) is equivalent to the condition
that the spectrum of T be contained in the closure of 8.
It follows from classical spectral theory that, for a normal operator T,
the spectrum itself is a spectral set. For non-normal T, the spectrum is
in general not large enough to be a spectral set. The introduction of the
concept of spectral sets is motivated just by the fact that one can build
up, for functions on a spectral set 8 of a not necessarily normal operator
T, a functional calculus which is analogous in some respects to the
functional calculus for normal operators. The map r(z) -> r(T) from
functions to operators extends namely, by continuity implied by (1),
to all 8-analytic functions, i.e. to those functions u(z) defined on 8, which
are uniform limits, on 8, of bounded sequences of rational functions.
These functions form an algebra, closed with respect to uniform convergence, and the map u(z) -> u(T) is an algebra homomorphism for
which also the metric relation (1) is valid. (It is just this metric relation that distinguishes this functional calculus essentially from the
functional calculus of Riesz-Dunford, based on contour integration in
the resolvent set.)
Some more properties follow easily: if 8' is the set of values of the
^-analytic function z' = u(z) taken on 8, then 8' is a spectral set of the
operator Tr = u(T), and for any ^'-analytic function v(z') the function
vou(z) = v(u(z)) is S-analytic, and we have vou(T) = v(Tr).
The usefulness of the theory depends on whether one can find spectral
sets for a given operator. It is obvious that any set containing a spectral
•f Read by Professor P. R. Halmos.
SPECTRAL SETS
413
set as a subset is itself a spectral set, but the meet of two spectral sets
need not be again a spectral set. Fundamental is the following theorem
of von Neumann[5] on contraction operators T, i.e. for which \\T\\ < 1:
Theorem 1. The unit disc 8Q = {z: \z\ < 1} is a spectral set for any
contraction operator T.
Using linear maps of 80 one gets as easy consequences criteria that the
set{z: \z — a\ < r},or{z: \z — a\ > r}, orRez > 0 be a spectral set of a given
operator T. These are, respectively, flî7 —a7|| < r, \\(T — al)-1]] < r~\
and Re T = |(JP + T*) ^ 0. It would be very desirable to obtain further
simple criteria for spectral sets.
The original proof of von Neumann of Theorem 1 uses a theorem of
Schur on analytic functions; a proof given by Heinz[4] uses the Poisson
integral formula; another proof, given by Sz.-Nagy[12] reduces the problem to the more simple case of unitary operators (see § 4).
The definition of spectral sets works also for Banach space operators.
But Foia§[1] has proved that Theorem 1 is valid only for Hilbert space.
For any spectral set 8 of the operator T, which is bounded, closed,
and whose boundary F is a Jordan curve (briefly: a bounded Jordan set),
the class of ^-analytic functions coincides, by a theorem of Walsh, with
the class 0(8) of functions which are continuous on 8, and holomorphic
in the interior of 8. In this case, a more refined functional calculus has
been built up by Foia§[2J. Essentially by taking the real parts of holomorphic functions and the corresponding operators, he extends the
map u(z) -> u(T) to the class sé(T) of (real or complex valued) functions
which are continuous on 8, and harmonic in the interior of S. Thus
extended, the map does not remain multiplicative, but it has the monotonic property: u(z) < v(z) implies u(T) < v(T); the metric property (1)
is also true. Since any continuous function on F can be continued
uniquely to a function in ^(T), so we have in reality a map of the continuous functions on F, to operators. Passing to limits by monotone
sequences, this map can be extended to broader classes of functions
on F, in particular to the characteristic functions co(ß, z) of Borei subsets
of F. The corresponding operators o)(ß; T) ^ 0 form the so-called harmonic spectral measure of T with respect to 8. The properties, and the
use in functional calculus, of this 'measure' are studied in detail in
Foia§'s paper[2].
2. Positive definite functions
There is still another possibility of building up a functional calculus for
general operators, with metric properties such as (1), namely by deducing
414
BÉLA SZ.-NAGY
it directly from the functional calculus for normal operators. This will
be treated in the following sections.
The fundamental concepts here are those of dilations and projections of
operators. If T is an operator on Hilbert space H, and T is an operator
on Hilbert space H containing H as a (not necessarily proper) subspace,
we say that T is a dilation of T or, equivalently, that T is a projection
of T, in signs
^
, m
^
m
&
T = pr# T or simply T = pr T,
if we have, for any h € H,
Th = PTA,
where P denotes the operator of orthogonal projection onto the subspaee H of H. (These terms are due to HalmosC3] and Sz.-Nagy [ul ,
respectively ; Halmos says ' compression ' instead of ' pro j ection '. )
An important pertaining theorem deals with operator-valued functions
T(s) on a group G, which are positive definite in the sense that, for any
Jî-vector-valued function h(s) such that h(s) ==
j 0 only for a finite set
of elements s e G, we have
2
S (T(1r*s)h(s),h(t))>0.
(2)
s€<? teG
Particular positive definite operator-valued functions on G are those for
which T(t~xs) = T*(t) T(s); in case T(e) = I (e denoting the unit element
of G) these operators T(s) are thus unitary and form a representation of
the group G (i.e. we have T(s)T(t) = T(st)). The theorem in question
reads as follows:
Theorem 2. For any positive definite H-operator-valued function T on
the group G, with T(e) = I, there exists a representation of G by unitary
operators XJ(s) on some Hilbert space H ( ^ H), such that
(a)
2 » = prU(s)
(seG),
(b) the elements of the form U(s) h (s e G, h e H) span H. The 'structure'
{H, U(s), H] is then determined up to isomorphism. If G is a topological
group, and T(s) is a weakly continuous function on G, then U(s) is weakly
(and so also strongly) continuous too.
This is a straightforward generalization of a theorem of Gelfand and
Raikov on complex scalar-valued positive definite functions on groups,
and was proved first by Neumark[7]. Later Sz.-Nagy found independently
essentially the same proofE13], and generalized it[14] to semi-groups G
with unit element c and with an involution s->s*\
(st)* = t*s*9
s** = s, e* = c.
SPECTRAL SETS
415
_1
The role played in a group by s is played here by s*; the operatorvalued function D(s) is a representation of G if it satisfies the conditions
D(e) = I,
D(st) = D(s)D(t),
D(s*) = [D(s)]*;
for a commutative semi-group G all D(s) are therefore normal operators.
Let JBA be a self-adjoint operator on Hilbert space H, which is a nondecreasing, right-continuous function of the real parameter À, with
Km Bx = 0, Km Bx = / ; call BA an operator distribution j-unction (o.d.f.).
A-> — oo
À—> + oo
Particular o.d.f.'s are the spectral families (called also 'resolutions of the
identity ') : this is the case if the operators Bx are (orthogonal) projections.
Now each o.d.f. Bx gives rise, by the formula
(T{s)f,g)=^_j^d(Bxf,g)
( — 00 < s < oo),
to an operator T(s), which is a weakly continuous positive definite
function on the additive group of reals. By Theorem 2, T(s) is the projection of a weakly continuous unitary representation XJ(s) of this group.
By Stone's theorem, we have
U(s) = f°° eisXdEx,
and conclude that Bx is the projection of the spectral family EA. This
proves the following very useful theorem of Neumark[6>7]:
Theorem 3. Any operator distribution function Bx on Hilbert space H
is the projection of some spectral family EA on a suitable Hilbert space
H(2#).
An alternative proof may be obtained from Sz.-Nagy's generalization
of Theorem 2 to semi-groups G with involution; it is namely easy to
see that (putting B_^ = 0, B+O0 = /) Bx is a positive definite function
on the extended system of reals À ( —oo ^ À ^ oo), made to be a semigroup with involution by defining
Ào/£ = min{À,/4> A* = À (e = +oo).
Further applications of Theorem 2 and of its generalization to semigroups are given in [14]: one obtains in this way a generalized form of
Neumark's theorem to additive operator-valued set functions[8]; a
characterization of 'subnormal' operators due to Halmos[3] (an operator
is subnormal if it has a normal extension on some larger Hilbert space);
a theorem on the operator moment problem; and theorems on unitary
dilations of contraction operators. The last-mentioned theorems are
important for our present subject, so we shall treat them in more detail.
416
BÉLA SZ.-NAGY
3. Unitary dilations of contraction operators
The starting-point here is the observation that, for any contraction
operator T on Hilbert space H, the operator T(n) defined by Tn for
n = 0,1,2,..., and by (T*)~n for n = — 1, — 2,..., is a positive definite
function on the additive group of integers n, i.e.
XZ(T(n-m)hn,hJ>0
n m
(hn — 0 for almost all n). This sum is namely equal to
lim
i-f"'(K(r,ftMft,m)d4>,
r - > l - 0 ^TTJO
where h(<fi) = yEle-in^>hn, and
n
K(r, <f>) = S r N e ^ T(n) = 2 Re (fJ + J «T'A
= Re(J + zT) ( I - z ï 7 ) - 1 ^ 0
(z = re**, 0 < r < 1);
the positivity of K(r, <fi) follows by the relation
{K(r,ftf,f)
= -B»{(I + zT)g,(I-zT)g)
=
lgl*-t*lTgl*>0,
for arbitrary / e H and g = (I — zT)~xf.
Thus, applying Theorem 2, we get the result (Sz.-Nagy[12'13]):
Theorem 4. For any contraction operator T on Hilbert space H there
exists a unitary operator U on some Hilbert space H ( ^ H) such that
y w = prU w
(?i = 0,l,2,...),
(3)
and H is spanned by the elements JJnh (n = 0, ±1, ±2, ...; heH). The
'structure9 {H, XJ,H} is then determined up to isomorphism.
If T, Tr are two contraction operators such that T is permutable with
T' and with T'*, thus also T* with T' (we say that T and T' are doubly
permutable), then T(n,m) = T(n)Tf(m) = Tf(m)T(n) is a positive
definite function on the additive group of vectors (n, m) with integer
components. This follows as above, making use now of double Fourier
series, and observing that the positive operator functions K(r,<ß),
K'(r', (j)'), attached to T and Tf, respectively, are permutable, and their
product is therefore also positive. Theorem 2 then implies that
T(n,m) = prUHJ'™
(n,m = 0, ± 1, ±2, ...),
and in particular
rpnrp'm =
pr
UHJ'm
(n, m = 0, 1, 2, ...),
(4)
SPECTRAL SETS
417
with permutable unitary U, IT. A similar result holds for any finite or
infinite set of doubly permutable contraction operators.
The problem whether simple permutability of T, T' suffices in order
that the representation (4) be possible with permutable U, TJ', is not yet
fully settled. The problem is equivalent to that of finding an operatorvalued function X(n, m) which is positive definite on the group of the
vectors (n, m), and reduces to TnT'm for n, m ^ 0. A necessary condition
for positive definiteness is that X( — n, —m) = X*(n, m), so one can dispose only of the values of X(n, —m) for n,m ^ 1. Brehmer, in his
Potsdam thesis, showed that the definition X(n, -m) = (T'*)mTn
meets the requirements under certain additional conditions, such as
the condition ||jP|| 2 +||r'|| 2 < 1, or that T be an arbitrary contraction
operator, while T' be an isometric operator. He obtained analogous
results for any finite or countable set of permutable contraction operators.
Theorem 4 has a continuous counterpart (Sz.-Nagy[12»13]):
Theorem 5. For any weakly continuous one-parameter semi-group T(s)
(s ^ 0) of contraction operators on Hilbert space H there exists a weakly
(thus also strongly) continuous one parameter group TJ(s) ( — oo < s < oo)
of unitary operators on a Hilbert space H ( ^ H) such that
T(s) = prU(s)
(s^O),
and H is spanned by the elements TJ(s)h ( — 00 < s < oo, heH); the
'structure' {H, U(s),H} is then defined up to isomorphism.
This follows from Theorem 2 if we remark that the operator-valued
function T(s), when continued to negative values of s by putting
T(s) = T*( — s), is a (weakly continuous) positive definite function on the
additive group of reals, i.e.
2S(?X-O^>Äm)^0
n m
for any finite set of real numbers sn and vectors hn e H. For commensurable sn this inequality follows from what has just been said for the
powers of a single contraction operator, and this implies the general
case by continuity.
The fact that any contraction operator has a unitary dilation was
proved in 1950 by Halmos[33 by a simple matrix construction. (For
isometric instead of unitary dilations this was proved already by Julia
in his Comptes Bendus Notes 1944.) However, the unitary operator
constructed by Halmos does not satisfy the relation simultaneously for
all n. After Sz.-Nagy published Theorem 4 in 1953, Schäffer[9] succeeded
27
TP
418
BÉLA SZ.-NAGY
in generalizing Halmos's matrix construction so as to yield a unitary U
satisfying (3) for all n. However, this construction seems inadequate to
be applied to the problem of two or more permutable T's or to one
parameter semi-groups. As a matter of fact, the U's attached by his
construction to different T's on H, are all defined on the same space H,
but are not necessarily permutable even if the T's are doubly permutable.
Call the operator U attached to the contraction operator T in the sense
of Theorem 4, and the semi-group V(s) attached to T(s) in the sense of
Theorem 5, the strong unitary dilations of T and of T(s), respectively.
There are some—rather loose—spectral relations between a contraction operator T and its strong unitary dilation U:
(i) If T itself is unitary, then U = T.
(ii) If T is not unitary, then the spectrum of U covers the whole unit
circle P 3 , Th. 2; or ™, cor. 2.2.).
(iii) T and U have the same eigenvalues on the unit circle, and the
corresponding eigenvectors are the same for both ([16], Th. 1).
(iv) The strong unitary dilations of proper contraction operators
(i.e. with \\T\\ < 1) are all unitarily equivalent to the orthogonal sum of
fa replicas of the unitary operator Uf(<j)) = e^f((f>) on L2(0,2n), where fa
denotes the dimension number of the Hilbert space H. (This has been
proved by Schreiber[10] with the restriction fa < J$0, and by Sz.-Nagy[15]
in the general case.)
Using the Hille-Yosida theorem in a form specialized to Hilbert
space one can characterize the one-parameter semi-groups T(s) of
Theorem 5 by the fact that their infinitesimal generator A satisfies the
condition: T = (A -hi) I (A —I) is a contraction operator not having the
eigenvalue 1 (see t1®^1®). Call T the infinitesimal cogenerator of the semigroup T(s). Any contraction operator not having the eigenvalue 1 is
the infinitesimal cogenerator of exactly one such semi-group. There is
a continuous analogue of (iv) :
(v) The strong unitary dilations V(s) of those one-parameter semigroups T(s) whose infinitesimal cogenerator T is a proper contraction
operator, are all unitarily equivalent to the orthogonal sum of fa
replicas of the one parameter unitary semi-group U(s)f(x) = eisxf(x)
on L2( — oo, oo);fais the dimension number of the space H (Sz.-Nagy[15]).
4. Functional calculus for general operators
If U is the strong unitary dilation of the contraction operator T, then
we have obviously
,-v
//m
/TT*
J
^(î7) = prw(U)
(5)
SPECTRAL SETS
419
z
for any function u(z) = cQ + c1z + c2z +..., holomorphic on a domain
containing the unit disc 80 in its interior. Thus we have
\\u(T)\\^\\u(V)\\
^suV \u(z)\,
(6)
the second inequaHty being a consequence of the spectral representation
of the functions of unitary operators, i.e. of the formula
^(U) = j o u(efi>)dEâ9
(1)
where Ee denotes the spectral family of U.
Applying this result in particular to rational functions having all
their singularities outside 80, we get a proof of Theorem 1 in § 1.
Moreover, these formulae enable us to derive a functional calculus for
T from that of U; namely we can define u(T) by (5) and (7) for any
function u(z) for which u(eie) is defined almost everywhere with respect
to the spectral measure generated by "E0, is bounded, and measurable
with respect to E#. However, in order to preserve also the multiplicative
property of the map u(z) ->• u(T), some restriction of the generality is
necessary: one possibility is to consider only those functions u(z) which
are bounded and holomorphic in the interior of 80 and whose radial
limits u(eie) = lim u(reid) exist almost everywhere with respect to E#.
r->l-
By virtue of (iii), § 3, these conditions are fulfilled in particular for the
functions, holomorphic and bounded in the interior of SQ, whose radial
limits exist at every point of the unit circle with the possible exception
of a finite or denumerable set of points, no one of which is an eigenvalue
of T. Denote the class of these functions by (PT(80).
The resulting functional calculus is studied in detail by Sz.-Nagy and
Foia§ in [16] . In particular, the metric relation (6) holds true for the class
@T(8Q), with sup \u(z)\ on the right-hand side.
1*1 <i
As an application they consider a contraction operator for which 1 is
not an eigenvalue, and the functions
m
u8(z) = exp I* —-^ I (s J* 0),
which belong to (PT(80); the operators Ts = us(T) then form exactly the
one-parameter semi-group of contraction operators whose infinitesimal
cogenerator is equal to the given operator T. This result opens a new
way to the study of such semi-groups. For example, assertion (v) in § 3,
on semi-groups, appears as a consequence of assertion (iv) on a single
27-2
420
BÉLA SZ.-NAGY
operator. The infinitesimal cogenerator of the strong unitary dilation
U(s) of T(s) is equal to the strong unitary dilation U of the infinitesimal
cogenerator T of T(s).
Independently, Schreiber[11] has proposed essentially the same
functional calculus. As definition he uses the formula
r2ir
^(T) = j o u(e™)dBe,
which results from (5) and (7) by putting Be = pr E0; Be is an operator
distribution function, uniquely determined by the equations
Tn=
\ "énedBe
(n = 0,1,2,...).
If T is normal, Bd can be calculated from the spectral family of T; in
this case—and only in this case—T is permutable with all Be's.
Sz.-Nagy and Foia§[16] carry over their results also to the more general
case when the role of a contraction operator and the unit disc 80 is given
to an arbitrary operator T on H, and to an arbitrary bounded Jordan
set S which is a spectral set of T (cf. § 1). Let z-^zQ = s(z) be a conformai
mapping of the interior of S on the interior of S0, continued to the
boundaries so as to be a homeomorphic mapping of S on S0. Let
-i
zQ->z = s (ZQ) be its inverse. Then s(z) is an ^-analytic function,
T0 = s(T) exists in the sense of § 1, and T0 has the spectral set 80 = s(8).
Thus T0 is a contraction operator (put r(z) = z in (1)). Let now U0 be
-i
the strong unitary dilation of T0 on the Hilbert space H. Then N = s (U0)
-i
is a normal operator on H whose spectrum is the image, by the map s,
of the spectrum of U0, and thus lies on the boundary of the set 8. We
have in particular
-i
-i
-i
-i
Tn = (s(T0))n = sn(T0) = prsw(U0) = [pr*(U 0 )] n = prN"
(n = 0,1,2,...), and the elements Nnh and N*nh (n = 0,1,2,...; h e H)
span H. This proves the following generalization of Theorem 4 (uniqueness may be proved directly, or by recursion to the uniqueness as
asserted in Theorem 4):
Theorem 6. If the bounded Jordan set 8 is a spectral set of the operator T
on Hilbert space H, then there exists a normal operator N on a Hilbert
space H ( ^ H) such that
(a) the spectrum of N lies on the boundary of 8,
(b) Tn = $TNn (n = 0,1,2,...),
SPECTRAL SETS
n
421
n
(c) H is spanned by the elements N h and N* h (n = 0,1,2,... ; h e H).
The 'structure' {H,N,JJT} is determined by these requirements up to isomorphism.
To get interesting applications one should remember, for example,
that any bounded Jordan set, containing the unit disk SQ as a subset,
is a spectral set of every contraction operator.
On the basis of Theorem 6 it is then possible to build up a functional
calculus for T with respect to 8 by deriving it from the functional calculus
for the normal operator N, in a similar way as it was done above for
contraction operators. An alternative way is to deduce this functional
calculus from that already existing for contraction operators, namely
by putting by definition
_x
u(T) = uos(T0)
-i
whenever uose 0Tçj(SQ).
The harmonic spectral measure of T with respect to 8 (cf. § 1) is
nothing else than the spectral measure of the normal operator N. Many
other results of the paper [2] appear thus in a more general setting.
REFERENCES
[1] Foia§, C. Sur certains théorèmes de J. von Neumann concernant les ensembles spectraux. Acta Sci. Math. 18, 15-20 (1957).
[2] Foia§, C. La mesure harmonique-spectrale et la théorie spectrale des
opérateurs généraux d'un espace de Hilbert. Bull. Soc. Math. Fr. 85,
263-282 (1958).
[3] Halmos, P . R. Normal dilations and extensions of operators. Summa
bras. math. 2, 125-134 (1950).
[4] Heinz, E. Ein v. Neumannseher Satz über beschränkte Operatoren im
Hilbertschen Raum. Göttinger Nachr. pp. 5-6 (1952).
[5] Neumann, J . von. Eine Spektraltheorie für allgemeine Operatoren eines
unitären Raumes. Math. Nachr. 4, 258-281 (1951).
[6] Neumark, M. Spectral functions of a symmetric operator. Bull. Acad.
Sci. Ü.B.S.S., Sér. Math., 4, 277-318 (1940). (Russian with English
summary.)
[7] Neumark, M. Positive definite operator functions on a commutative group.
Bull. Acad. Sci. U.B.S.S., Sér. Math. 7, 237-244 (1943). (Russian with
English summary.)
[8] Neumark, M. On a representation of additive operator set functions.
C.B. (Dokl.) Akad. Sci. U.B.S.S. 41, 359-361 (1943).
[9] Schäffer, J . J . On unitary dilations of contractions. Proc. Amer. Math. Soc.
6, 322 (1955).
[10] Schreiber, M. Unitary dilations of operators. Duke Math. J. 23, 579-594
(1956).
[11] Schreiber, M. A functional calculus for general operators in Hilbert space.
Trans. Amer. Math. Soc. 87, 108-118 (1958).
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BÉLA SZ.-NAGY
[12] Sz.-Nagy, B. Sur les contractions de l'espace de Hilbert. Acta Sci. Math.
15, 87-92 (1953).
[13] Sz.-Nagy, B. Transformations de l'espace de Hilbert, fonctions de type
positif sur un groupe. Acta Sci. Math. 15, 104-114 (1954).
[14] Sz.-Nagy, B. Prolongements des transformations de l'espace de Hilbert
qui sortent de cet espace. Appendice au livre 'Leçons d'analyse fonctionnelle* par F. Biesz et B. Sz.-Nagy. Budapest, 1955.
[15] Sz.-Nagy, B. Sur les contractions de l'espace de Hilbert. I I . Acta Sci. Math.
18, 1-14 (1957).
[16] Sz.-Nagy, B. and Foias, C. Sur les contractions de l'espace de Hilbert. I I I .
Acta Sci. Math. 19, 26-45 (1958).