69
Internat. J. Math. & Math. Sci.
VOL. 14 NO.
(1991) 69-76
CERTAIN INVARIANT SUBSPACES FOR OPERATORS WITH RICH EIGENVAUJES
KARIM SEDDIGHI
Department of Mathematics
Shiraz University
Shiraz, Iran
(Received September 29, 1987 and in revised form June 7, 1988)
ABSTRACT.
For a connected open subset
of
the plane
and n a positive integer,
let B () be the space introduced by Cowen and Douglas. In this article we study the
n
spectrum of restrictions of T in order to obtain more information about the invariant
subspaces of T.
When n=l and T e
Bl(fl)
such that o(T)
is a spectral set for T we
use the functional calculus we have developed for such operators to give some Infinite
dimensional cyclic invariant subspaces for T.
KEY WORDS AND PHRASES.
Invariant subspace, generalized Bergman kernel, zero sequence.
1980 AMS SUBJECT CLASSIFICATION CODES. 47B37, 47A25.
INTRODUCTION.
of the plane and n a positive integer, let
For a connected open subset
B () denote the operators T defined on the Hilbert space H which satisfy
n
(a)
(b)
(c)
(d)
ran(T-u)
H for
H,
ker(T-o)
dim ker(T-)
in
,
and
n for
in
.
The space B () has been introduced by Cowen and Douglas [I]. This class of operators
n
was further studied by Curto and Sallnas [2].
In partlcular, they show that an
,
be
the
realized as
adJolnt T-z of the operator of
n
multiplication by z acting on a Hilbert space K of coanalytlc functions on
having a
K
operator
T
in B () can
generalized Bergman kernel K.
To give an example, let A
the Hilbert space of analytic functions on the open unit disk D such that
The class B () is a rich class of operators.
2
denote
27
rdrdO<.
0
The operator S on A
2
0
defined by (Sf) (z)
easy to see that S* is in
z f(z) is called the Bergman shift.
It is
BI(D).
Shelley Walsh [3] has obtained several results concerning the backward Bergman
shift S*.
In particular, the spectrum of restrictions S*
where m is an invariant
K. SEDDIGHI
70
It Is also pointed out how this study sheds more light on
subspace for S, Is studied.
the structure of the Invariant subspaces for the Bergman shift.
purpose
The
the
of
present
is
article
to
results
these
extend
to
the
in hope that more insight into the Invarlant subspace structure for these
BIfR)
class
operators can be gained from this study.
2.
PRELIMINARIES.
Let H be a separable Hilbert space and let B(H) denote the algebra of all bounded
For T c B(H) the spectrum, point spectrum, compression
linear operators on H.
Oap(T)
o(T),op (T),
For the appropriate definitions see Halmo [4].
and approximate point spectrum of T are denoted by
spectrum
respectively.
and
If f c H, then [f] will
Let T be a bounded linear operator on a Hllbert space H.
be the smallest invarlant subspace for T containing f.
oc(T),
The notation
[f],
is used to
If A H, let VA denote
It is the smallest closed linear subspace of H that
denote the smallest Invarlant subspace for T* containing f.
the closed linear span of A.
contains A.
A subset S of the open unit disk D is said to be dominating (or dominating
for 8 D) if
Equivalently,
D is dominating if and only if almost every point
a subset S
of 8 D Is a nontangentlal limit point of S.
The following presentation on the generalized Bergman kernels is taken from Curio
and Salinas [2] and will be needed in the sequel.
For
set
A,
n
<
2
let
of
subspace
n
some
F(, n2)
.
Given a
C and
(R)}.
a
=I ,ak
k
on A will be any Hilbert space which Is a linear
{ak}kffil:
space
the
<2
n
{
n
a functional Hilbert
linear
space
2
all
of
n
-valued
B(
By a kernel functions on A we mean a function K:A x A
k
on T
Z(,)
C
n
If
^);
> 1, and every collectlon
Z(,) (all
for every integer k, k
k operator matrix
2
k.
in
addition
{K(i’ J))
Z
the k
t2)
n
that satisfies the
,
following conditions:
(a)
(b)
on A, for
functions
(I
k operator
i,
{1"**’k
k) is
J
matrix K
K in
a
(b)
is
A, the
positive
operator
inJective for every
I, we shall say that K is strictly positive.
collectlon {I’
^, all k
2
and
Let K be a functlonal Hilbert space on A. If for every
K
K and
K(k,.)
K
’k
^
<f(),
<f,K(,.)>
then we shall say that K
K
reproducln kernel for KK
>,
for every f
E
n’
K,
has a reproducing kernel on A and K will be called the
It is well known that Z Is unique.
71
INVARIANT SUBSPACES FOR OPERATORS WITH RICH EIGENVALUES
Note that if K is a strictly positive kernel function on A, it gives rise to a
([2] section 3). To see
functional Hilbert space on A with reproducing kernel K.
this let D(A) be the linear space of F(A,
ri=Ik K(Ri,.)i,
f
for
If g is another function in D(A), g
is
<’’’>K
is
Now
<K(li,j)i,j>.
consisting of all functions of the form
I e 2n
e A,
K( lj
)
j’
(i=l"’’’k)’ k
let <f
)
I.
k
’g>K-- ri,j=l
(A). It is easy to show that a
polntwlse convergent, and hence the completion KK of (A)
Cauchy sequence in D(A)
with respect to
Ri
k
El=
In2)
<’’’>K is
an inner product on
a linear subspace of F(A,
,2).n
Furthermore, by construction
it follows that K is the reproducing kernel for K
of
K. We also observe that if K
is a strictly positive kernel function which is already a reproducing kernel for a
KK,
K.
A, then KK
functional Hi[bert space K on
Let A be a domain in the plane and assume that every functional Hilbert space
on A is Invarlant under the map of multiplication by the conjugate of the function
C defined by f(z)
z. The restriction of this map to such a functional
f:A
Hilbert space will be denoted by T
the
in
z
denote the set of
2
Now let A(fl,
n
Note
plane.
that
n2)
(fi,
12-valued
n
is
coanalytlc functions on the domain
invariant
under
multiplication
functional Hllbert spaces on R under consideration will be invarlant under
functional Hllbert space on R which is a linear subspace of
a
conanaltic
functional Hllbert
A kernel function K:R
fi+
space
(R,
12n
will
by z.
All
T_.
b
A
called
on R.
2
B(I n is called sesqulanalytlc if K(.,.) is analytic
In the first variable and coanalytlc in the second variable. Given a sesqulanalytlc
positive operator matrix
K) be the (p+l) x (p+l)
kernel function K on R, let
Hp(l;
whose m, n-entry is
n
m
m!
z
m
8n
n!
K(l,l), 0
m, n
We say that the kernel function K is
every I e
a
(
p.
nondeenerate
if H (; K) is
p
inJectlve for
and every positive integer p.
Then there exists
Let K be a nondegenerate sesqulanalytlc kernel function on ft.
such that the nondegenerate kernel
coanalytlc functional Hilbert space K on
K
function K on fi is a reproducing kernel for
and denote by
the form f
in D
0
DIO
K
([2] section 4).
the linear subspace of A(,
l= 0
j, je
j
2
n
Indeed, let
10
e
spanned by functions of
r.q
f g
kK(10’"
k=O
k!
nk
is another function
we define
<f’ g>
P
J-EO
q
j
k--EO <
It follows that the completion
k K(XO, 0
j! k!
KI0 ,DAO
Klo.
of
coanalytlc functional Hilbert space on
a reproducing kernel.
Therefore K
K
J’
nk >
with respect
to this
which is independent of
inner product
0
is
a
and has K as
K. SEDDIGHI
72
A nondegenerate sesqulanalytlc kernel function K on fl Is called a generalized
Bergman kernel (g.B.k for brevlty) if T_ E B(KK) and for every k E fl, T*_ k has
Z
Z
range,
closed
ran K(4,.)
ker(T*
4) and dim ker(T*
Z
Z
4)
n, where
n
is
a
fixed
For a detailed treatment of
generalized Bergman kernels the reader is referred to Curio and Sallnas [2]. In
particular, a g.B.k. K on fl is strictly positive and hence D(R) is dense in KK.
Also K(4,4) is invertible for 4
The operator T*_ acting on KK is called the
positive integer throughout the
rest of this article
.
For simplicity of
canonclal model associated with K.
ntation
set T
T
Z
Finally we make a few observations which we will need later
IK(4,
)
(Differentiate
-
K and it is easy to see that
K
is in
the
multiplication by
i-IK(’’)
(T-4),.iK(4’’)
i.v
.
(i-l).V
We also use the fact that
is an inJectlve operator on K to deduce that o (T*)
K
P
0 it follows that ran(T- ) is dense in H. But ran(T- ),
(T-4)K(,.)= 0 i times).
relation
-)
Since ker(T*
Note that
X
R, is closed, so ran(T- )
3.
SOME INFINITE DIMENSIONAL CYCLIC INVARIANT SUBSPACES.
H.
.
In this section we will consider a speclal case.
Let T
In [5,6] we have proved the following There is
L, the interior of L is simply connected and L is
denotes the conformal map
minimal with respect to these properties
Moreover, if
0
from L onto D then A
(R)- is a spectral set for
(T) is in BI(()) and o(A)
A. Also Let(T)
Let(A).
a spectral set for T and o(T)
a compact set L such that o(T)
To study the invariant subspaces for such operators we may assume, without loss
D
of generality, that R is an open connected subset of the unit disk such that L
and T
section
In this case we also
is a spectral set for T.
We assume this normalization is in effect in this
KK.
then for convenience we let H
such that o(T)
BI()is
have D
O=
[5]).
(Seddlghi
If K is a g.B.k, on
In the proof of the next theorem we use the functional calculus developed in
6 ).
(Seddlghi
THEOREM I.
If f is in H then
[f]
PROOF
g
If g
{g
H: g
h(T)f
for every h in
h(T)f for every h in H
then g
p(T)f for every polynomial p.
Hence
[f].
Conversely suppose g
{pn
If] and h
H
There is a uniformly bounded sequence
of polynomlals converging polntwise to h in D.
Since
converges to h weakHence
h(T)
([6]).
weak-star
Pn(T)f h(T)f weakly,
Pn(T)/
so h(T)f
[f]. But g If], so g h(T)f and the conclusion follows immedlately.
COROLLARY I. If {Am} is a Blaschke sequence of distinct points in R which has all
its limit points on D and {Cm}m=l is a sequence of nonzero complex numbers such
star
it
that
Era=
follows
c
m
KAm is
[f]
that
in H then
{g
g(m
0
for all m}
Pn
73
INVARIANT SUBSPACES FOR OPERATORS WITH RICH EIGENVALUES
PROOF.
If gkk
.
if] and h is in H
Era=| c
m
h( )g(X
m
that h(X
have g(X
then h(T)f
0 by Theorem
m
h(k)=
and
m
<n g,f>
<T *n g,f>
Tnf>
<g
If g
0 for al[ m then
m
0 for k
Let
g()
0.
m
Since
m
such
0, we
[Ck}k=
kffil [Ck[ [[KXkI[
<
be a sequence of
kl Ck Kk’
If f
then if]
0 for all k}.
If g()t
k)
if] then
kl lckl
Since
g(
function In H
let h be a
be a sequence of distinct points in fl which has all its
[Xk}k;
0 for all k, then for any m we have
proof of Corollary I, so g
If g
m
ill
([6]) and <g,h(t)f>
h(m)K k m
Fix m and
m. Then
nonzero complex numbers such that
PROOF.
I.
m
so g e
0.
on @D and is not a dominating sequence, and let
[imit points
H:
c
lm=
Xm g (xm)
0.
m
THEOREM 2.
{g
__m__El Cm
kfEl
c
k
0
lust
as in the
[f].
k g(k
k
lK%kllk)<-,
al [I] we have c
Tmf>
<g,
g(X
0, for any m.
sumk ICk gk)l
the
|__
Is finite.
By Theorem 3 of Brown et
0, it follows that
Because ck
0 for all k.
Note that for any k,
g()-
0 for
all k.
4.
THE SPECTRUM OF RESTRICTIONS OF T.
PROOF.
oap(T[m f Op(Tjm).
Let
Then there
Oap(TIm)
.
.
and T
In this section we assume that K is a g.B.k, on
model In B () acting on H
KK. We also assume that o (T)
n
p
I.
LEMMA
Let m be an Invarlant subspace for T then
T* Is the canonical
exists a sequence of unit vectors
{fk}
in
H with
[[(T )fk[[
where
(T
k
L)gk
2
0.
n and gk ran K(L,.).
0.
Because T
complement of ker(T
l kl
k).
Wrlte
Since
fk K(,.)
(T-
L)fk
(T-
+ gk’
)gk’
it follows that
has closed range, It is bounded below on the orthogonal
From this observation we get that gk
/
l kl 12,
O.
which we conclude that
K. SEDDIGHI
74
Therefore
{k
then
a
jZI=
is uniformly bounded in norm by a constant C.
[ak l,j}i=!
subsequence
{aj}j=
l.n
Let
all
C for
k_j
Then
For
k.
such that a
2
in
ki
Hence
.n
[akj }nj=l’
k
n there
IIK(X,.)ki- K(X,.)II
o.
Ii
We have shown that
+
k gki
K(k,.)
fkl
K(I,.)
K(k,.)ki
in H.
Observe that
Therefore K(k,.) is in m. Since
K(k,.)
sequence of unit vectors, K(k,.)
:
h(k)
2
in
<h, K(,.)>
<h(k),>
>
0, then h()
Hence ker(T- k)
0}
o
p
so
(T[ m
o(TIm
Let m
p
[f],.
then since {: f()ffi 0}
have
oc (Tlm)
o
T Im )" We
To prove o (T
(T
m
0.
). To see this assume
0,
2
in
n
).
op(TIm
now apply Lemma
{: f(%)
<g, (T
%)h
>
<(T*
K(,.)>
2
for every
in
particular,
<pn()f(X),
Hence
%)g,h
>,
{: f(k)
m
0}. If we prove
to complete the proof.
Pn(T*)f
<(T*
>--pn(%)
m"
so (T*
0
If g
((T-)m) then for any h in m
k)g
(m)
%)g.
(T*
)g, K(,.)>
pn(k)f()ffi (Pn(T*)f)
f(X)
(T
0} it suffices to show that if f(k)
of polynomials such that
<Pn(T*)f,
c
), by the argument preceding the theorem, we
is not in the compression spectrum of T
{pn
p (Tlm
Hence h(k)
).
We first show that
this
sequence
2.n
This says that <h, K(,.)>
0 for every h in m.
m
in
If m is a cyclic invariant subspace for T* then
THEOREM 3.
0
0 and observe that
0.
0 for every
We also need the fact that {: f()
then
0}
f(k)
Then approximating h by polynomials in T* applied to f, we have
E m.
PROOF.
:
we have
<p(X)f(X),
<p(T*)f, K(X,.)>
f()
for T* and discuss the
To see this let f(k)
0 for every h in m
for every polynomial p and any
Note let h
--[f],
Before doing this note that {
spectrum of the restriction T
is a
(fki)
).
o (T
m
p
0, so
We now consider a cyclic invariant subspace m
k
convergent
a
is
in C.
aj
kl,
J,
each
If
m. Thus
there
a
2
In
It follows that
0
() converges weakly to zero in
o.
is
Because f(%) O,
75
INVARIANT SUBSPACES FOR OPERATORS WITH RICH EIGENVALUES
we have
(T*
follows that T*
qn(T*)
Therefore
-
COROLLARY 2.
g, so g
f
then f()
{: f()
If],.
Kk >
O}
op(Tlm ).
(m)
so
is not in the
[k
Op(Tlm)
{k: f()
0}. Indeed if
Kk
e m
By the argument preceding Theorem 3 we have
Op(Tlm
Thus
)--{%:f()-0}
{:h()
and f in H the notation
in
0 for every h in
f({k})
A sequence
f({%k})
Zero sequences for A
occurs in
{k
}.
0 is used to mean that
N times then f has a zero at k of order at least N.
let n(%) denote the number of times
exists f in H with
k)m
Thus ((T
then
If n--
O.
[k
DEFINITION I.
H.
0 for every h in m}.
{: h(k)
For a sequence
)
ran(T
it
and m is a cyclic Invarlant subspace for T* then
<f,
occurs in
-)
closed,
m
If n
Let m--
) is
)g. Since ran(T*
(T*
(m)
m
o(Tlm
PROOF.
Then
%
is bounded below on ker(T*
compression spectrum of T
if
pn(k)
pn(Z)
pn(k)/ O. Let qn(Z)
z
)qn(T*)f Pn(T*)f -pn()f
We also
{}.
of points in
if there
is a zero sequence for
0.
2
are studied by C. Horowitz [8].
He has shown that they are
quite different from Blaschke sequences.
THEOREM 4.
m
{h:
Let n
h({kk})
I, and
be a zero sequence for H.
{k
If
O} then
(Tlm
0 for every h e m}.
{:h()
It is easy to see that
PROOF.
m--V{IKx:
e:
{Xk},
i
-O,...,n(k)- I}
Therefore the finite linear combinations of functions of the form
N
If %
{}
n
k=IZ if ioZk ck’i iKk’
n(kk)
where n
k
for each k are dense in m.
then
N
Z
k)
(T
C
i=O
k=l
N
n
r.
E
k
k=l i:0
Ck’t
iK
((T- k
k i
BiKk
+ (X
k
X)
k
=k=Ir
N
Ck, 0 (k-
)
+
Kkk
k=Ir i-IZ
KXk
i-IKkk (k
iK+kE Ck ’nk (kk
Ck, i (i
nk-I
k--I i--I
)i
+
)
N
[(i+l)Ck
i+l
+
(kk- k)Ck
i
-k)
i
K
nk Kkk"
K. SEDDIGHI
76
These functions are dense in m
so
is not in the compression spectrum of T
m
He n c e
Note that k
h
E
p (Tlm
m}. Applying Lemma
ACKNOWLEDGEMENT.
The
if and only if K
x
Op(T Im
so
e m
{X:h(X)
0 for every
research
council,
we obtain the result.
author
would
like
to
thank
the
Shiraz
University, for a grant (No. 65-SC-392-187) during the preparation of this article.
The content of this article Is presented to the ICM 86, Berkeley, August 3, 1986.
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3.
COWEN, M.J. and DOUGLAS, R.G., Complex geometry and operator theory, Acta.
Math. 141 (1978), 187-261.
CURTO, R. and SALINAS, N., Generalized Bergman kernels and the CowenDouglas theory Amer. J. Math. 106 (1984), 447-488.
WALSH, S., Cyclic vectors for the backward Bergman Shift, Dissertation,
4.
HALMOS, P., A Hilbert space.problem book, Van Nostrand, Princeton, N.J. 1967.
5.
SEDDIGHI, K., Yon Neumann Operators in
I.
2.,
University of California, Berkeley, 1984.
156.
6.
7.
8.
BI(R),
Canad. Math. Bull. 27 (1984), 146-
SEDDIGHI, K., Weak-star closed algebras and generalized Bergman kernels, Proc.
Amer. Math. Soc. 90 (1984), 233-239.
BROWN, L, SHIELDS, A.L., and ZELLER, K., On absolutely convergent exponential
sums Trans. Amer. Math. Soc. 96 (1960), 162-183.
HOROWITZ, C., Zeros of functions in the Bergman spaces, Duke J. Math. 4__[1
(1974), 693-710.