Internat. J. Math. & Math. Sci.
Vol. 8 No. 4 (1985) 669-675
669
DIAGONALIZATION OF A SELF-ADJOINT OPERATOR
ACTING ON A HILBERT MODULE
PARFENY P. SAWOROTNOW
Department of Mathematics
The Catholic University of America
Washington, D.C. 20064
(Received January 18, 1984)
ABSTRACT.
For each bounded self-adjoint operator T on a Hilbert module H over an
H*-algebra A there exists a locally compact spaceTand a certain A-valued measure
2
such that H is isomorphic to L ()A and T corresponds to a multiplication with a
continuous function.
operators.
There is a similar result for a commuting family of normal
A consequence for this result is a representation theorem for generalized
stationary processes.
KEY WORDS AND PHRASES.
H*-algebra, Hilbert module, A-linear operator.
MATHEMATICS SUBJECT CLASSIFICATION CODE. PRIMARY: 46H25.
1980
SECONDARY: 46K15, 47BI0,
47A67, 46GI0, 60Gl0.
|.
INTRODUCTION.
The diagonalizatlon theorem states that for each bounded self-adjolnt linear
operator T acting on a Hilbert space H there exists a measure space
valued measurable function h(s) such that H is isomorphic to
responds to the multiplication with h(s).
(S, ) and
L2(S,
a real
and T cor-
Furthermore, the space (S, ) could be
selected in such a way that there is a Hausdorff topology on S with respect to which
h(s) is continuous, S is locally compact and which makes
a regular Borel measure.
In this note we shall give a suitable generalization of this fact.
The situation is somewhat more complex in our case.
be replaced by the tensor product L
2
()A,
L2(S,
The space
which is less manageable.
needs to
This space is
properly defined below.
2.
PRELIMINARIES.
Let A be a proper H*-algebra (Ambrose [I])and let rA
xy[x, yA be
its
trace-class (Saworotnow and Friedell [2]); let X be a locally compact Hausdorff space
and let
be a positive
is defined on the class
A-valued Borel measure on X.
for some compact set Q, and
Members
fflwill
The last statement means that
of all Borel subsets / of X having the property
is such that
(P()x, x) >
be called bounded Borel sets
0 for all
"and
(a bounded Borel set is a Borel set
included in a compact set).
Note that the scalar-valued function
an ordinary Borel measure on
X;
m= trt, is
I.[ (Definition
it coincides with the total variation
in 111.1.4 of Dunford and Schwartz
[3]) of.
that/Q
each xA.
P.P. SAWOROTNOW
670
Let S(X) and S(X,A) be respectively the classes of all complex-valued and
Ei ibAi(x)
and
(x)
Eia
(Ai , aiA
q)(x)
One can define the integrals for members
A-valued simple functions of X.
[(x)
and S(X,A) in the usual way by setting
jd
ii
and
and
A ’s
=
are complex numbers) of S(X)
(2. ;)
ai
and then extending it to larger classes using the norms
]=fi i m
X
(2.2)
and
Let L(X) and B(X,A) denote respectively the classes of those functions to which the
(Note that S(X) is dense in L(X) and
integrals are extendable in this fashion.
S(X,A)
B(X,A)).
is dense in
Then it is easy to se that
hold for
ruer
all,eL(X)
.
and
(X,A).
(For a discussion
o
tE
a ana eithereL(X)
f
or
eB(X,A),
then
"(x) >
0" holds o,,tide
FepFesented in the
o
0
or
or
each
o
some set
=i’]laii
or
"
which
aria
/a#
O.
The first assertion is easy to veriy.
PROOF.
a
this type we
aeS(X,)
0
that
o
integrals
the reader to Bogdanowicz [4 ).
with
ArB
with
Let
8
ill’ fi2’’’"
i square -easurabIe coplex-valued unctlons.
trflA=
O.
disjoint
tr i) >
r(fli)
be a simple function such
0" holds.
Then
(ACB)
can be
and
Then
Then there 18 a r-valued
nner
produc
(I’ 2
() such that
defined on
2
2
tr[2’
2i
is an ordinary scalar
product on L () making L () a Hilbert space.
LE 2.
Le t
l’
n
’2
L2 ()
trEi,ja
PROOF.
Let
n()
*
i
jdaj >
denote the nor on
tr
tr(aja
and let a l,a 2,...,an
a
()
A.
Then
(2.6)
0
n(O) 2
2d.
(,)
Let
>
0 be
daj
iOj-ij )du)
i
’(ajai)’
Oj-i"j )d") !
and the lst su can be made arbitrarily small by selecting
small enough.
On the
other hand one can see that
tr(t,ja
jdua 3)
(ja33)(tat)*d
0
(2.7)
671
DIAGONALIZATION OF A SELF-ADJOINT OPERATOR
(
since
ajDj)(Y iaii)*
COROLLARY.
is positive and
Note that the expression (za,a)
PROOF.
Hence (za,a)
>
A.
al,a 2,...,an
tr(a*za) is of the same form as trz.
Define the positive form
[f,g]
(here g
jj,bj).
= {fK:
Let
It is not d[fficult to see that
ni=liaK
If,g] on
i
with
on L
(2.8)
i,jai ijd)b.
If,f]
L2()A
0,., ..K’
K’we
[s a Hilbert
module.
define L
then T
atl
2()A
to be
Define the operator T
h
(A by setting
Th(ia [)
Th(f)
glso
1,2,...,n L2()
by setting
Let h be a bounded continuous real valued function on X.
2
tA.
is a
0 for each aA.
Now consider the space K of all tensors
and
tra i
daj _> O.
poitlve member of
Hence
simple.,
Ei,J (a]ri:d:’a’)IJr
TJ
3
The expression z
T
h
h
Z(@ih)a
is a bounded seif-adjo[nt (in the sense that
s A-Inear
fL2(A,
(2.9)
[Th(f),g]
[f,Th(g)]
(additive and A-homogeneous in the sense that
Th(fa)
hoIds),
Th(f)
for
a
*s bounded (in the sense that
h
some M) can be verified directly, using [0 of Naimark [5].
The fact that T
fixed member of K.
tr[f,Ty(f)]
p(y)
on the space
n
subsection 4 of
p(e),
3.
tra
be a
(2.10)
daj
BC(X) of all bounded continuous (complex) functions on X.
from the proposition I
g-linear
E{iai
Let f
Consider the positive linear functional
0
in Nalmark
It follows
[5] that p(h*h)
<
Thus
operator ts of the for T
h
described above.
IN RESULTS.
An
Definition.
if there exists f
.
(we assume that
THEOREM
Ho
g-iinear
operator T on a Hilbert oduIe H is said to be cycIic
such that the
T(f O
If
O
set{:=0"kTk(fo)ak:ak’A’k complex}s
dense
n
H
f ).
O
For each bounded A-linear self-adjoint operator T on a Hilbert
module H there exists a locally compact Hausdorff space X, a tA-valued positive
regular measure
defined on the class
of bounded (dominated by compact sets) Borel
subsets of X and a bounded continuous real valued function h on X such that H
2
isometrically isomorphic to L (PA and T corresponds to the operator T h (described
2
above) acting on L ()A. If T is cyclic, then X is homeomorphlc to the compact
subset of the real line.
PROOF.
operator
Let B be the commutative B*-algebra generated by T and the identity
(note that each member of B is A-linear).
ideals of B, let
be the standard Gelfand topology
Letbe the
on, and
on
Celfand map of B [nto the continuous complex functions
set of maximal
let S>S(M) be the
Note that is
homeomorphic to the spectrum of T, which is a compact subset of the real llne.
consider 2 cases.
We
672
P.P. SAWOROTNOW
CASE [.
H such that the
First assume that there exists
n
tt
i=ISi (fo)ai:siB’ai
(3.|)
is dense in H (this is equivalent to the statement that T is
Let
-->P
[5])
section 4 of Naimark
(17,
be a spectral measure on
such that S
,[nce it coutes with linear
cyclic).
ofTL(eachA’is
be the class of all Borel subsets
compact) and let
set
=S(M)dPM.
bounded
Note that each
is a rA-valued positive measure
onfl,
can take X
(3.2)
o
<-->
In this case we
S(M)
(3.4)
This correspondence can be extended in the obvious way to the isomorphism
between the closure of K and the Hilbert space
also preserved by this correspondence: if
extend
L2(NA
L2(N).
SI,S2B
[fo’SS2f o]
[Sfo’S2fol
The rA-valued inner product is
then
g(M)S2(M)d(M)
this isomorphism to a correspondence between H
(].5)
and a dense subset of
by setting
EkSk(o)a k <-->
Sk(Mak
(].6)
This correspondence also preserves the (vector) inner product: if f
g
(J 3).
(linear)
C().
We
Sf
[fo ]’
o
The correspondence
Sf
is a
o
denotes the generalized inner product on H).
=.
Then
and for each SeB we have
f
above,
is A-][near
[fo ’Pf o
o’fS(M)dPM]
PM
(M)d(M) =S(M)d[fo if’
as
P
maps f-->fa(a6A) (which commute with all SB).
A--> A:
(here,
since’bi
Proposition II [n sub-
=Qi fo )hi’
Sk(f o )a k
[f’g]
Ek,iak[Sk (fo)’Qi(fo)]bt .iak gk(M)Qi(M)db
L2(NA. It Is easy to check
We extend it to an isomorphism between H and
correponds to the operator T h of multiplication with function h(M)
k
T(E kSk(fo)ak
Ts" k (fo)ak
case is
Note also that in this
compact subset of the real line.
CASE 11.
and
then
coincides with the set H(f)
T(M):
(]" a)
homeomorphic to the spectrum of T, which is a
This implies the last assertion of the theorem.
i=iSi(f)ai:SiB,ai
Saworotnow [6] that fH(f).
that T
<--> E kT(M)Sk(M)ak
Now let us consider the general case.
closure of the set
(
For any fH let H(f) be the
Then it follows from Lena 2
in
Also both H(f) and its orthogonal complement H(f) (which
p
gH:[g,h]
0 for all
hH(f) (Lemma 3
of Saworotnow
[6])) are invariant under T.
It follows from this fact and Zorn’s Principle that there exists a set
#,
and each
For each
H(f) is invariant
yFand SB let S
exists a compact Hausdorff space
under T.
be the restriction of S to
(,
),
H(f),
and let
B
a rA-val,,ed positive Borel measure
g
and
673
DIAGONLIZATION OF A SELF-ADJOINT OPERATOR
2
on’FE>, such
by(
a continuous real valued function
that
H(fy)
is isomorphic to
(>)A
and action of the operator T F (the restriction of T) corresponds to the
m,[tiplicat[on with hy on L-(>,). Note also that hy(M)
T
for each M
IJ
__<
>,
Let X
be the topology on X defined by the requiremet that a
OX is open (O ) if and only if Of]?7>,belongs to r) for
Let
be the
U
set
and let
,
each>,.
class of all bounded Borel subsets of X.
simplified notation here) [,2
Thenis
on
a ri,g and
g is a positive
X by setting h(M)
by(M)
where
For
eachthere
such that
/i(Jl 71’’I
rA-valued measure
yFis
such that
are indices (we use a
We set
on.
We define the function h
M?7y.
Then it is easy to see
that h has the req,[red properties.
To complete tle proof
F[t note that each
er[fy).
Now let
f-g
that
<
L2()
with
>
For each
i(L-().
e j=l j
But each
i
g
s
=
g
[=I
can be approximated in
L2(j
j
with
0 one can find
be approximated (as close as we please) by
i.e.,
L2()A =yL2(gy)A.
L2()
L2 (gy) (easy to
is now sufficient to show that
is included in L-(g) and that
fL2(A.
,.xpress[ons of the form
’an
[t
for some
ebers_iln
oiai
such
L() by
YI’ Y2’’’’’Yn
Thus
a menber of
Conversely, let
fL2(g)A
expressions of the type
1.2(g)-- withieS.
_n
_@a
i=li
hea
can be approxiated by
with a.,A and
We may conclude that
fCL2(gA
01,02
since L
"n belonging to some
2(g)L2(g) for each
.
The
reader should be able to give a precise argument here.
THEOREM 2. Let Z be a family of bounded A-linear operators on a Hi[bert module
H (over an H*-algebra A) such that each member of Z and its adjoint (with respect to
the generalized
product)
commute with any other member of Z.
In particular, Z
:ould be a commutatve *-algebra of A-linear operators on H.
Then there exists a
locally compact Hausdorff space X, a rA-valued positive Borel
measureon
"T >
||
in,er
h
T
X and a map
L2(g)A
of Z nto complex valued functions on X such that H is isomorphic to
and each T crrespnds t m’itiplicatin with sme functin
II
PROOF.
Mreever
hT"
The proof is essentially the same as the proof of Theorem
llhTl
above.
<We
the *-algebra of operators generated by Z (and the identity operator I) instead
of the algebra generated by the operator T (and I).
COROLLARY
|.
A-linear operators
Each *-representation of a commutative *-algebra by bounded
s
of the form
with a complex valued function h
.
x--> Th,
h
x
where T is an operator of multiplication
h
described before Theorem
This corollary could be considered as a generalization of Theorem 65 in Mackey
[7] if
we disregard the fact that Mackey considers more general
(self-adjoint)
algebras and we do not specify the space X on which the functions h
h
x
act
(also
our Hilbert module does not have to be separable (as a Hilbert space)).
COROLLARY 2.
and let
-->U
Hi,bert module
Let G be a commutative locally compact group with composition +
be a *-representation of G by A-linear unitary operators acting on a
H.
Assume that there exists a vector
generated by the vectors of the form
compact Hausdorff
spaceZ,
U%(fo)
foH
such that the submodule
is dense in H.
Then there exists a
a positive rA-valued Borel measure
on’fl,
and a map
HO,
P.P. SAWOROTNOW
674
of G into the continuous functions on such that H is (isometrically)
2
isomorphic to L ()A and each U t corresponds to multiplication members f
-->gt
L2()
gt"
t-->gt
The map
has the following properties (for each tG and all
M:
(3.10)
(here 0 is the identify of G)
g0(M)
with
Igt(M)
(3. ll)
(3.2)
gt(M)
gt(M)gs(M)
g-t(m)
gt+s (M)
(3.13)
It is appropriate at this point to mention a crtain application of the last
be a generalized stationary
corollary. Let G, A and H be as above, and let
’G-->H
is an H-valued function on G such that
pcocess (Saworotnow [8]), i.e.,
((t),(s))
((t+r),(s+r))
t. vectors of the form
(t),
(H#
tG
closure of
For each tCG consider the operator U
ut(E nk= |(t k)ak)
Then the map
t-->U t
iq
H
on
E k=|.(tkn+t)a k
members of
arrived at a
H
is isomorphic to L
L2 (/)
with
k=|(tk)ak:tkG
and let
(.|4)
2
Let
C() corresponding to
()A and each U
gt(M)f(M).
ht(M)
on.
defined
Z’,
and F,
fO =(1)).
corresponds to m. ltiplication
In ths fashion we
representation of the abstract stationary process
co,crete
generated by
).
f(0).
f0
2 are fulfilled.
Coro|lary
For each tG let
gt"
-omplex valued continuous f,nction h
((t),(s))
submodule
defined by
he as in Corol|ary 2 and let f(M) be the member of
o
be the
a representation of G by A-linear unitary operators and
t is easy to see that the ,sumptions of
Then the space
H_
Let
for all t,r,sG.
by the
Note that the scalar product
corresponds to the expression
t(M)gs(M)f(M)f(M)dg(M)
gt_s(M) f(M) 2d,(M)
/gt (M)g_s(M)I f(M)12d,(M)
d/(M)
t(M)
f
and this expression depends on t-s only and is independent
.
o
(3.|5)
a particu|ar choice of
and s.
CONCLUDING REMARK.
To conclude the paper we make the following remark about the operator Th
d[scussed above.
It [s easy to see that we do not need at all to assume existence of
(locally compact) topology on the space X (discussed
paper). Let be a positive tA-valued measure defined
a
X.
If h is any
tr-measurable
at the beginning of this
on some o-rlng of subsets of
essentially bounded real valued function on X then
2
the corresponding operator T on L
h
()A,
Th(Eii (R)al)
Ei(ih)(R)al
(3.16)
The fact that T is bounded can be
h
verified in the same way as above using the algebra B of all essentially bounded
is also self-adjoint, A-linear and bounded.
tr-measurable complex-valued
functions on X.
DIAGONALIZATION OF A SELF-ADJOINT OPERATOR
675
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AMBROSE, W. Structure Theorem for a Special Class of Banach Algebras, Trans.
Amer. Math. Soc. 57 (1945), 364-386.
2.
SAWOROTNOW, P.P. and FRIEDELL, J.C. Trace-class for an Arbitrary ll*-algebra,
Proc. Amer. Mth. Soc. 26 (1970), 95-100.
DUNFORD, N. and SCHWARTZ, .J.
Linear
Operators, Part I, Intercience, New York,
1958.
4.
BOGDANOWICZ, W.M.
5.
NAIMARK, M.A.
6.
SAWOROTNOW, P.F.
191-197.
7.
MACKEY, G.W.
8.
SAWOROTNOW, P.P. Abstract Stationary Processes, Proc. Amer. Math. Soc. 40
(1973), 585-589.
HALMOS, P.R. Measure Theory, Van Nostrand, New York, 1950.
9.
A Generalization of the Lebesgue-Bochner-Stieltjes Integral
and the New Approach to the Theory of Intergration, Poc. of the National
Academy of Sciences 53 (1965), 492-498.
Normed Rings, Moscow, 1968.
A Generalized
Hi[bert
Space, Duke Math. Journal 35 (1968),
Commutative Banach Algebras, Lecture Notes
University, Livraria Castelo, Rio de Janiero, 1959.
|0.
ROBERTSON, A.P. and ROBERTSON, W.J.
University Press, 1966.
11.
SAWOROTNOW, P.P. On a Realizatio of
Soc. 15 (1964), 964-966.
2.
SAWOROTNOW, P.P.
1952, Harvard
Topological Vector Spaces, Cambridge
a
Complemented Algebra, Proc. Amer. Math.
Linear Spaces with an H*-algebra Valued Inner Product, Trans.
Amer. Math. Soc. 262 (1980), 543-549.