IC/66/12
INTERNATIONAL ATOMIC ENERGY AGENCY
INTERNATIONAL CENTRE FOR THEORETICAL
PHYSICS
GENERAL EIGENFUNCTION EXPANSIONS
AND GROUP REPRESENTATIONS
(Lecture Notes)
K. MAURIN
1966
PIAZZA OBERDAN
TRIESTE
IC/66/12
INTERNATIONAL ATOMIC ENERGY AGENCY
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
GENERAL EIGENF UNCTION EXPANSIONS
AND GROUP REPRESENTATIONS
K. MAURIN*
(Notes of three lectures given at the ICTP 7-10 February 1966)
TRIESTE
February 1966
* University of Warsaw, Poland,
GENERAL EIGENFUNCTION EXPANSIONS
AND GROUP REPRESENTATIONS
In this paper I shall give some results concerning a general theory
of eigenfunction expansions for families of commuting self-adjoint opera•
tors in a separable Hilbert space! H (§2). As was remarked (in a special
case) by Gelfand and Kostiucenko in the case of an arbitrary spectrum
the (generalized) eigenvectors are elements of <£>^ - the dual of $ ,
where | is a dense linear subset of H equipped with such nuclear topology that the identical imbedding
i: <f ^ H
is continuous.
Thus we have always to do with a triplet of
(0A)
# c H c $'
(locally convex) vector spaces, each dense in the following one, but only
H is a Hilbert space.
The most classical example of the triplet (0.1)
is
LzOt) c St'OR?)
where 2)CiR?)
(o.2)
is the L. Schwartz space of infinitely differentiate
functions with compact supports and S^CVt)
butions on H, .
the space of all distri-
,
In § 1 we recall the fundamental notion of the Hilbert-Schmidt (H-S)
mapping, and give the definition of a nuclear space by means of H-S
mappings, which, as was shown by A, Pietsch,is equivalent to the
original (much more involved) one by Grothendieck.
ity of Q)0C) - or more generally QC&h) - follows.
-1-
A proof of the nuclear-
In § 3 it is shown how the fundamental theorem of § 2 can be applied
to obtain a decomposition of a unitary representation of a locally compact
(1. c.) group into irreducible (or factor) representations.
The first step
in this direction has been taken by Mautner and V. Neumann.
In § 4 it
is shown that the irreducible spaces H(X)are common generalized eigenspaces of some self-adjoint operators constructed in a natural way.
SI.
HILBERT-SCHMIDT (H-S) MAPS.
NUCLEARITY
All Hilbert spaces considered here a r e separable, i. e.,they have a
countable orthonormal basis.
Let
Hu . • K. = 1,2 be Hilbert spaces with a scalar product (•!•)
and let (,6-j)
De a n
orthonormal basis of H^ .
tinuous mapping /\; H1 —» HZf
the number
F o r any linear conIAl. — 21. HA&jH^ i s
independent of the basis ( £: ) .
Definition 1.
If | A | <oo then the map
A e L (ll x
1
H^)
is called an H-S map.
Lemma 1.
The identical operator I « L (H , H)
is H-S if and
only if the dimension of H: dim H is finite.
Proof:
III - ZLhejti
« Dim H,
Let oL be an open subset of iK. or (more generally) a differentiate
(separable) manifold of dimension n .
finite differentiable functions on Si
C^" iSV)
is the set of all in-
with compact supports.
be a precompact open subset of Si .
H
~
h (-~y*)
Hilbert space of measurable (complex) functions on Si
Let ^>;
is the
with the scalar
product
The derivations a r e understood in the sense of the theory of distributions.
-2-
Ho
is the closure of
C^XSli)
in the I
fl-norm.
Theorem L (cf MAURIN [6] )
1°
If Si* is a pre-compact domain of an n-dimengional differentiable
manifold, and if /* ^> «/2.
2°
,
U^
o
then the canonical imbeddings
If <5"l^ possesses a regular boundary then the imbeddings
H-S.
Corollary.
Let SI, s*'SI i, e.
S2t ^Slz C • • - ; , be a sequence of
pre-compact domains exhausting
^ r » : ^r /y
t»,j
(-%•/))
'
Then to each
e A/such that the imbedding
/**'
v
"J
The spaces
Q(J2) ,
Sh and let
*», / e / /
there exist such
.#*
i s of H
~* "/
^(J2J) ,
nj •' - ^ // / W ^/2.: J
9/(Jl)
, ^(Sl)
"S
type
*
, are defined in the
following way:
Definition 2.
&(&)
is the set G°° (SX) with (locally convex) topo-
logy defined by the norms I
l\m ' , t",j - Wr-f where
L
,
Slj/JZ
We introduce now the notion of nuclearity.
Definition 3.
A locally-convex vector space
exists an equivalent system of seminorms
$
is nuclear if there
( / U )*£ J?
1° For every /?e B is the (normed) quotient space
Xfi:= i>/ty , where /£ =
a prehilbert. space.
-3-
such that:
2° F o r each/? £ " O there exists such p£ o that the imbedding
A Hilbert space H is nuclear if and only if dim H < aa ,
Proposition 1.
Proof:
The topology of H is given by one norm
IV = {of and -3f - H .
is H-S.
\\
If
only.
Hence
In virtue of 2° the identical imbedding H -^>H
Thus Proposition 1 follows from Lemma 1,
Proposition 2.
As was proved by Grothendieck,nuclearity is preserved
by the most important operations on locally-convex vector spaces:
a)
taking the quotient by a closed subspace
~Af C <p
(i. e.f if $ is nuclear)then 5 ^ / " is nuclear too;
b)
the completion
$ of a nuclear $
ia nuclear;
C) the strict inductive limit (J = liMyf^* of nuclear spaces $[ is
nuclear,
i. e . , §. C $„ C . . .
i w\ *" ZK
, § = \J $,
then the topology induced on 0 ^ by
the topology of f K is identical to that of
of
<p^ .
The topology
^ is - by definition - the finest topology on $> such that
for each i
d)
, and if
the imbedding
the (topological) direct sum
$. —> ffl is continuous;
<§) §{ of nuclear spaces is nuclear.
t
Definition 4.
&0Z)
the following topology:
^ftj
/* St.
.
<fy -precompact.
is the open space
CT (SI)
4>: .'= { <P € C™(fl)
provided with
: the support of f c
$ ; has the (nuclear) topology of the space 0 (j^j)
^ ) (JZ) = r i t ^ (f.. ,.
We can give now a simple proof of the following fundamental result of
Grothendieck.
Theorem 2. (Grothendieck}
1° ^ (51) is nuclear;
2° 0 (JL) is nuclear.
-4-
^j
,
Proof: 1° follows immediately from Definition 1 and the Corollary,
2° follows from 1° and Proposition 2c,
If
$ is a locally-convex vector space {1. c. v. s.) then _>
| denotes
the dual space of $> , i. e, <§''= /_ f«£ / C) * i . e . , the elements of
are linear continuous functionals on $
Definition 5.
on
SI
and
Elements of
au(&) and €>(-%/&re called distributions
distributions with compact supports in SL respectively. It
can be proved that. ££)' and %,' are nuclear.
§ 2.
GENERAL EIGENFUNCTION EXPANSIONS
Let us recall rapidly the notion of a direct integral of Hilbert spaces.
Let (A,cr) be a locally-compact (I.e.) separable measure space.
each Xe A
there exists a Hilbert space ( H (*V ,
be a vector field on A
, i. e., h()
(• I- ) x ) .
is a map: A * A —>
For
Let
A(A) e H
A countable family of vector fields is called a fundamental family if:
1°
all functions
A * X —=> (A-M I A:(*)) € €
are
O"-measurable for all i, j = 1, 2 , . . .
2°
for each Ae A
the set
{ A: A i - / - ^ ?f •• • 1
space HfAV
A vector field n() i s
Definition 6.
The direct integral H of Hilbert spaces
spans the
cr-measurable if each function
is the Hilbert space of equivalence classes of cr-measurable vector
fields At) for which
The scalar product on H is defined by
•= f
H is denoted by
/
//A)^.
A
Diagonal and Decomposable Operators.
JV —*• A tA) £• L-(rlM)
/A 3 A —> (A(\)L(x)
An operator field
i s measurable if all functions
, y--<*-?re cr-measurable where { Aj 0 ) )
\ Lt(\))
is the fundamental family.
A measurable operator field (A ( • ? ) is
called a decomposable operator in the Hilbert space H = j . Hf>-)ofa^,
The simplest decomposable operators are diagonal operators: if T &
j_ (A^) i . e . , if j - is an essentially bounded cr-measurable function then
X ~> J}(\) ;= /A)Tfa
)
called diagonal operator.
~f~ >
> where I (X) is the identity in fi fA) and is
Of course, one could introduce diagonal operators
\ _ ^ gfa) Jh \
with unbounded measurable
a
, but
since such operators are not bounded their domain is not the whole space
If &t" is a weakly closed algebra in L ( H) then at
denotes the
commutant of jt. It is the set of all bounded operators in H commuting
with every operator in (ft
.
A weakly closed * -algebra in L ( H) is
called a Von Neumann (y. ]NQ algebra.
The following fundamental result
due to von Neumann gives a nice characterization of algebras of diagonal
and decomposable operators.
Lemma 2. (v. Neumann)
1°
The algebra -v/) of diagonal operators is a commutative v. J\|,
algebra.
2°
of the
The map
* -algebra
j _ (G^J ^f ~^> If € ^
L°° (ff) onto /vA (both algebras are provided with weak
topologies): T^j- = /7 '7J
3°
is a topological isomorphism
etc
The commutant r3^
of the
diagoi
the algebra n^i
n5l of diagonal
operators
is the v. N. algebra J£ of (all) decomposable operators: $)=- ui. and the
commutant Ji
of ^
is the algebra
-6-
Thus an operator is decomposable if and only if it ia commuting with all
diagonal operators.
In the spectral theory of unbounded self-ad joint ( s . a . ) operators
the notion of strong commutativity is of paramount importance.
Definition 7.
Two s . a . operators Aj, , ^
their spectral families
{ £"/A)}
,
a r e strongly commuting if
K^^z
a r e commuting.
This is equivalent to the commutativity of Cayley transforms of At
Aj. : (Ak* iX)(Ak~>I
and
J i *•-//?.
We can now formulate the principal result of this section.
Theorem 3. (Nuclear Spectral Theorem cf MAURIN [2, 6] )
Let Aj Af B be a denumerable family of strongly commuting s . a .
operators haying a common invariant domain 1)
1° there exists such a nuclear space that
dense in H :
$ ( ^ H C f is a Gel' fand
triplet;
2°
each Af
maps
3°
there exists a direct integral H - JA HMtrfcy
Hilbert isomorphism
$
9^:
continuously into <J> : A^
H
: $ —•> $ ;
and such a
> H that for & -almost all Afjf(X)C <§>
and Hf A)are common generalized eigenspaces of all A^ : if e (A1) €. H (A),
then
^
identically for f f %(
(2.1)
where \_A^}
: XfAj
is the spectrum of A^
(2.1) can be written shorter
A^ €&) ~ %J\)
the natural extension of Af
given by the identity
-7-
e(^)
* where A\
is
or plainly fl^ -> Ap - / | ^ ,
Taking in each
H (X) the orthonormal basis
<?k(x),.. k - -f, ?, - -,
we obtain the generalized Fourier-Plancherel equation
,
4
r
f £ -
(2.2)
the isomorphism jr : H-> H is given by
where
J^A) -" Z
5°
<^^^;>^/AJ,
the spectral synthesis of J'6'f
(2.3)
is given by
P ~ /$&)</<>*
(2.4)
where 5^ is given by (2. 3)
In view of the applications to the theory of unitary group representations the following corollary is of great importance (cf § 3.),
consider H as a subspace of
Corollary.
that *Ut U
1L:- (1/*)
f
so we have the following:
Let LL be such a unitary decomposable operator in
i <p
We can
H C Q)
> ty are continuous (cf Remark 2° below); and let
t>e t n e natural extension of "U to $>' (precisely
<$ := (U¥/^)/
) and let 1(&):= U IHM,
then
(This integral is not a symbolic way of writing a decomposable operator
but a genuine strong integral).
-8-
Remarks.
1°: The isomorphism Jr diagonalizes simultaneously all operators
,
i.e.
and is called thefAp)-Fourier transform.
We obtain the classical Fourier
transformation taking ff- L7(Vt*)f Ar ~>'£ , <fi ~ 0(1%.h) (<*
Then Awhere
IH" ^ 1Hh
, and
eM - efX-. )
€(*, *) =• e*f>(-i(\tx,+ - •-, >„><* )J , Jk - />,,... / A* ) ^ x - rxy/
2°: vonNeumann proved "only" that to every abelian # <—• v. N-
algebra there exists an isomorphism j
taneously all operators Ap € vTT
for cT-almost all Xe/\ .
t^l
\H
^ H diagonalizing simul-
, i. e., for which {*) is satisfied
In this way we can identify <fb with the algebra
of diagonal operators and hence the commutant at of xJb with the
algebra of decomposable operators.
§ 3.
UNITARY REPRESENTATIONS OF LOCALLY-COMPACT GROUPS
Let
Gr be a separable I.e. unimodular group, i . e . , a group withabi-
invariant Haar measure.
On every 1. c. group one can introduce nuclear
function spaces e>(G) &{(?) respectively which in the case of a Lie group G"
are simply the (infinitely) differentiate functions and regular functions
with compact supports respectively (See Appendix),
£)'€&)
.
The dual spaces
resp- 2J> CG)are called spaces of distributions resp. distribut-
ions with compact supports in Gr . Clearly we have inclusions
9CG } C l}(e) C£fre) ,
£>CG) C gfe)C
&C6) .
One can define in
the usual way the convolution of distributions
Since in the case of a Lie group the invariant differential operators
-9-
can be given by distributions with support in 6 :
In that way, the invariant enveloping algebra of any 1. c. group Gr can
be defined as an algebra C^e °f distributions with carriers in (e \ ,
the multiplication in %># being the convolution. We have at our .disposal a much 'bigger algebra: the whole space ^ CG) . The unit
in & is the Dirac measure c£ . This algebra is an involution
algebra.
Definition.
Let
<p (§)'—
VCf')
,» then for each
1~ €
T~*'t 2)'(G) is defined by
9
t
T±
(3.1)
Let *U = 0Ut H )
be a unitary representation of a 1. c. group Ox ;
£2 a —^ %{„ € L(H) . Our aim is to construct to each unitary representation ( % 14 - ) a Gel'fand triplet $ c H C ^
and a set (1/(T))
s
of,operators in H such that
1° lia acts continuously in
§ ; V* 1 §
>$
;
2° the operators 1t(T) are mapping continuously © into itself
3° the operators UU) are symmetric:
'M'T) C 14.(T)
anc j
their closures are self-adjoint;
4°
u (fk ) are strongly commuting,
k s 1,2. ^
5° K(T)
are commuting with the operators Ua 4 & G .
vId
Applying our fundamental Theorem 3 we obtain thus a decomposition of
the representation (Vs Mj into a direct integral of irreducible representations / (%lM MM) do"
an(
^ ^ n e irreducible spaces H (V) are common
-10-
s
(generalized) eigenspaces of the operators V.(Tj.
The Girding space H <y of the representation '% ")
set of linear combinations of vectors
9f. * 9(G)t 4h& H
.
1((9?)hk
is defined as the
•' — Jr 9"f<i)&Q>'hkd4
Plainly H& is dense in H .
We can now transplant the representation
*t
to a
#•-representation
of the algebra £? (GJ , whence to the enveloping Lie algebra
the group
&?g~ of
Gr in the following way.
Theorem 4.
(cf MAURIN [4] )
For each
l€
W(G)
and <&€ MQ
(3.2)
or, more precisely, for every ~^V/£ , KUtr)^
(>'>;-
Then 1° W$ 7~-> HCf) is a -X -representation of the algebra & &>')
by operators in H& : U(T*S}n - 2/fT)Ufr)a , a ? HQ ; U(Tf) CU(rf
Thus for symmetric operators T = T4" the operators are symmetric.
2°
The representation ( *U(T) ) is an extension of the represent-
ation {1(, f4 ) since
1A' CcL )'- Ua
/
7 '
Let Z(c$~) denote the centre of an algebra <Jv
Theorem 5. The construction of a Gel'fand triplet adapted to ( ^U.
Let
Wo C H
be any Gel'fand pair we can provide the set of
linear combinations <& > — ZaiMfy}*,'
. 9" € 0C(S ) >f - & U
with such nuclear topology that <£ c H C (^'
is such a Gel'fand
triplet with all desired properties 1° - 5° .
Proof:
Let us prove, for example, that
-11-
H ).
taking T - o* we obtain by virtue of Theorem 4. 2°
From the continuity of the maps
the continuity of the operators
J0^ $2? —•> 7~* ^ f 0
1/Cr) ; |>
follows
> (Jj
In order to obtain a decomposition of { 1f(•// ) into a direct integral
of irreducible representations JA (i(.Mf MMJcfcr
we take any
maximal abelian algebra in 11 - the commutant of the v. N. algebra generated by the operators
families of
WT)
(
$£
ae C
T=7+cZ(%J
Then the irreducible spaces
H IX\ ^ $
which contains all spectral
<n Tf
Z.(%'fe)).
can be considered as common
generalized eigenspaces of the operators *7>t(T).
The most important models of unitary representations are provided
by shift-operators on a homogeneous space
(where ^
are functions on the space
measure ,<x on X, we take
distributions on the space
lief)
l~€r ^ '
a
,X= Gr/H
X ).
{Z/a /)(*)
•~
If we take an invariant
H » L ( X, u),our construction leads to
X ':
(§> • r
^fX)
The operators
r e then invariant "differential" operators (they are dif-
ferential operators in case of differentiate homogeneous space X )
the irreducible spaces H (X\ are eigenspaces of the "differential operators"
tlCf)
'
We want to stress once more that our whole theory does not
suppose either the differentiability or the compactness of the group.
If
<q- is compact and we take for l€ ~Z(& (&)) a function, we obtain integral convolution operators from the theory of Weyl and Peter.
ACKNOWLEDGMENTS
This paper contains the material of three lectures delivered by the author at the International Centre
for Theoretical Physics, Trieste (7-10 February 1966). He would like to thank Professors Abdus Salam and
Paolo Budini and the IAEA for the hospitality extended to him at the ICTP.
-12-
APPENDIX
Distributions on 1. c. groups
1.
The spaces Qfc}, gfe)
1. c. group.
, &?&J, S?fej.
Let Gr be a separable
We define here only the distribution spaces for the most
important limit cases: a) G-connected,
b) G-discrete.
a) Let G be a connected separable group.
A famous theorem of
Yamabe a s s e r t s that to each neighbourhood CL of the neutral element
C of G there exists such 4 (compact) normal subgroup 7*K C &
that Gt: - Gi
ftk
is a Lie group.
Q) (G^) a r e nuclear.
with a subspace
Taking
2) ($^
.
We know (cf % 1) that the spaces
n^n,}
•
Identifying
we can identify
QC^i^
2)C6h ) with §K
)
- a space
of continuous functions on G which a r e constant on such left coset
^^
-it.)
we obtain an increasing sequence of nuclear spaces © C d> C •" C
Let
Q)CC) '• = itV $K .
Then
0
(G)(as a strict inductive limit of
nuclear spaces^is nuclear (cf §1).
b) G- is discrete.
Let K-^G be an increasing sequence of com-
pact subsets of G exhausting G .
spaces
Since the sets K • a r e finite the
C ( K ; ) a r e of finite dimension and hence they a r e nuclear. <$ is now
defined as a strict inductive limit of C(K\):
2.
Nuclear topology on
Let
HD C H
<2) (G) x Ho
&CGr) J : = ^
<$ can be introduced in the following way.
be a Gel'fand pair, whence the (topological) product
is nuclear.
Consider the mapping
Let N be the kernel of T :
/V '• = T
fo)-
linear isomorphism [T] of ( 0(G) x H o ) / N onto
closed, the space \2)CG)X //O )//V
by [T]
C { K^) ,
7
T
:
S)(G) X He, ^
Then T induces a
^ .
Since N is
is nuclear ( proposition 2a ) carrying
that nuclear topology on <£> - the image of [T] - we obtain the
desired nuclear space
0 .
* Let TTK; Cr-* <S/ K K = & K where ~H*<<£) = V ^ K ,
/ ' € C C^At )
. T ^ is continuous. If
then 7T*; Cfek ) -* C6j) is given by rf: /4> /oTk
-13-
BIBLIOGRAPHY AND REFERENCES
A.
Books
All proofs of the theorems mentioned can be found in the author 1 s
monograph: "General Eigenfunction Expansions and Unitary Representations of Topological Groups", Warsaw (Monografie
Matematyczne),in
the P r e s s .
Von Neumann and Mauther
Theorems a r e presented in monographs
of J. Dixmier (r T authier-Villars, P a r i s , 1957 and 1964) and Naimark
"Normed Rings" (Noordhof, Groeningen, 1964).
B.
Papers
[1]
F . BRUHAT, Bull. Soc. Math, de France 89, 43-75 (1961).
[2]
K. MAURIN, Bull. Acad. Pol. Sci., Sgr. math. ast. phys.
7., 471-479 (1959).
[3]
K. MAURIN, ibid. J3, 845-850 (1961).
[4]
K. MAURIN, Math. Ann., in print (1966).
[5]
K. MAURIN, L. MAURIN, Bull. Acad. Pol. Sci. lJJ, 199-203
(1965).
[6]
K. MAURIN, Math. Scand. ^, 359-371 (1961).
-14-
•^TMBrwrwjrTr-i.T "•
-BtmHIfcBTr?,'" * m
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