On the spectral maximal spaces of a multiplication operator

ON THE SPECTRAL MAXIMAL SPACES OF A MULTIPLICATION OPERATOR

JAE CHUL RHO AND JONG-KWANG

Yoo

1.

Introduction

In [13], Ptak and Vrbova proved that if T is a bounded normal op- erator T on a complex Hilbert space H, then the ranges of the spectral projections can be represented in the form

£(F)H = n (T - >"1)H for all closed subsets F of C,

>.f£F

where £ denotes the spectral measure associated with

The alge- braic representation of the spaces £( F)H turned out to be useful for the automatic continuity of linear transformations intertwining a given pair of certain decomposable operators. For applications in automatic continuity theory, we refer to [3], [8],[11],[12],[18].

In the present paper, we show that a theorem describing structure of certain class of invariant subspaces of multiplication operators, so called spectral maximal spaces(terms to be defined below). And we attempt to give a simple and unified approach to the algebraic repre- sentation of the spectral maximal spaces which is an analogy of the normal operators, i.e., if A is a semi-simple commutative Banach alge- bra and Ta has the weak-2 spectral decomposition property then there exists an idempotent element rEA such that

Tr(A) = n (Ta - >"I)A for all closed subsets F of C,

>.f£F

Received January 9, 1995.

1991 AMS Subject Classification: 47All, 47B40.

Key words and phrases: Decomposable operator, Divisible subspaces, Spectral maximal spaces, Hull-Kernel topology, Gelfand topology.

This work was supported by KOSEF Grant 931-0100-005-1.

where Trx

rx for all x

A.

We first recall some definitions and known results concerning local spectral theory included in [5]. Given a complex Banach space X,

£( X) denotes the Banach algebra of all bounded linear operators on X. Given an operator T

£(X), let Lat(T) stands for the collection of all closed linear subspaces of X which are invariant under T. Recall that the notion of local spectrum from [5]; if x

X then the local spectrum O'T( x) of T at x is defined to be the complement of the set of all

C for which there is analytic f : U

X on some open neighborhood U of

such that (T - p.1)f(p.) = x for all

U. A spectral maximal space of T is an invariant subspace Y of T, which contains any invariant subspace Z with the property O'(TIZ)

O'(TIY).

An operator T

£(X) is called decomposable (resp, weak-2 spectral decomposition property) if for every open cover {U, V} of C, there exists Y, Z

Lat(T) such that X = Y + ^{Z (resp, X} = Y + Z ), O'(TIY)

U and O'(TIZ)

v.

It

follows from the example given by E. Albrecht [2] that in gen- eral, weak-2 spectral decomposition property is strictly weaker than decomposability. An important feature in this theory of the decom- posable operator is the investigation of the spectral maximal spaces given by X T( F)

{x EX: O'T( x)

F} for all closed subset F of C. If T

£(X) has the single-valued extension property, then for each closed F

C, XT(F) is a linear subspace (not necessarily closed) of X, hyperinvariant for T. Let T be a linear operator on a complex Banach space X. We say that a subspace Y of X is T-divisible if (T-.H)Y

Y for each

C.

Consider the class of all linear subspaces Y in X which satisfy (T-

~I)Y=

Y for all

C \ F and set E _{T (F)}

spanY. It is obvious that

(T-~I)ET(F)=

ET(F) for

C\F as well so that the class which we

consider has a maximal element if ordered by inclusion. In particular,

ET( if» is the largest T -divisible subspace for the operator T. The

study connections between the ETU and XTU subspaces derives from

automatic continuity theory. The maximal algebraic spectral subspaces

ET( .) are defined in algebraic tenus and the local analytic spectral

subspaces X

are analytically defined. Their structure is much more

readily accessible to analysis, see [5], [10],[11], [12], [14), [15), [18J for

details.

In [8], B. Johnson and Sinclair were the first to investigate the re- lationship between automatic continuity question and the existence of divisible subspaces, i.e., for T

LeX) and S

£(Y), if T is not al- gebraic and S has a non-trivial S-divisible subspace, then there is a discontinuous linear operators A : X

Y satisfying AT = SA.

In order to have more information about the splitting spaces Rho and Yoo [16] have introduced the following concept of decomposability.

An operator T

LeX) is called (E)-super-decomposable if for every open cover {U, V} of C, there exists an R

LeX) commuting with T such that R

= R, u(TIR(X))

U and u(TI(I-R)(X))

V. The class of (E)-super-decomposable operators contains all spectral operators, all normal operators on Hilbert spaces, all operators with a totally disconnected spectrum. For various examples and characterizations of (E)-super-decomposable operators, see [16].

2. Maximal spectral spaces of a multiplication operators Given a commutative complex Banach algebra A, let 6(A) stands for the character space of A, i.e., the set of all non-trivial multiplicative linear functionals on A. For each a

A, let a : ^6(A)

^{C denote}

the corresponding Gelfand transform given by a( tP) : = tP(a) for all tP

.6.(A). A commutative Banach algebra A is said to be regular if for every closed subset

6(A) and any tPo

K, there exists x

A such that x( tPo)

and x( tP) =

for all tP

K. Recall that if A is semi-simple then a

a is injective, equivalently, the radical rad(A) of A is {O}. On 6(A) we shall have to consider the hull-kernel topology, which is determined by the Kuratouski closure operation

cl(B) = ^{tP

^{6(A) : tP(u)} =

^A with t/J(u) =

for each t/J

B}

for B

6(A). In fact, cl(B) = hul(Ker(B)). The hull-kernel topology is coarser than the Gelfand topology on 6(A) and they coincide if and only if the algebra A is regular, see [4]. For a E A the multiplication operator Ta

£(A) is defined by TaX

for all

A.

THEOREM

1. Let T

LeX) be (E)-super-decomposable. HT has

no non-trivial divisible subspaces, then for each closed F

C there

exists an idempotent operator R

.c(X) such that the ranges of the R coincides with XT(F) = ET(F).

Proof. Let U be an open neighborhood of F. Then there exists an idempotent R

.c(X) such that TR = RT, u(TIR(X))

U and u(TI(I - R)(X))

C \ F. Since TR = RT, we have (T - >')RET(F) = RET(F) for all >.

C \ F, which implies RET(F)

ET(F) by maxi- mality. A similar argument ensures that (1 - R)ET(F)

ET(F). We infer from u(TIU -R)(X))

C\F that (1 -R)ET(F)

(1 -R)(X)

ET(C \ F). And so (1 - R)ET(F)

ET(F) n ET(C \ F) = {O}, since T has no non-trivial divisible subspaces. Hence ET(F) = RET(F)

R(X). On the other hand, it follows from u(TIR(X))

U that R(X)

XT(U) n ^ET(U). Hence we have R(X) ~ n ^{(XT(U) n ET(U»}

F<;'U

= XT(F)nET(F) ~ ^XT(F) ~ ^ET(F) ~ ^R(X), because XT(F) = ^{n{XT(G) : G open, F} c G}. Hence R(X) - ET(F) = XT(F), which completes the proof.

We denote by cOO( C) the Frecht algebra of all infinitely differ- entiable complex valued functions defined on the complex plane C with the topology of uniform convergence of every derivative on each compact subset of C. An operator T

.c( X) is called a generalized scalar operator if there exists a continuous algebra homomorphism .p : COO (C)

.c(X) satisfying .p(I) = 1 and .p(z) = T, where 1 is the identity operator on X and z is the identity function on C. For a given generalized scalar operator T

.c(X) and a closed subset F of C, P.

Vrbova proved [19] that the existence of a natural number nE N such that XT(F) = n ^{(T - >.)n X.} From this equality, we have

>o.~F

ET(F) ~ n ^{(T - >.)m X ~} n ^{(T - >.)n X = XT(F).}

>o.~F,mEN >o.~F

Hence X T( F) = ET(F) for a closed F

C, and so every general-

ized scalar operators do not have non-trivial divisible subspaces, since

ET(<!J) = ^XT(<!J) = ^{O}. On the other hand, the Volterra operator T defined on C([O, 1]), the set of all continuous functions with the supre- mum norm, given by (Tx)(t):= Lt x(t)dt for all x

C([O,l]), and t E [0,1]. Then {f E COO([O,l]) : f(n)(o)

0 for all

E N} is a non-trivial T-divisible subspace and T is quasi-nilpotent.

COROLLARY

2. Let T

£(H) be a normal operator in Hilbert space H and let EO be its spectral measure. Then for each closed F

C, there exists an idempotent operator R

£( H) such that

R(H)

HT(F)

ET(F) = E(F)H

n ^{(T - A)H.}

>..fJ.F

Proof. If Z

H is aT-divisible subspace then by Lemma 5.1 of [18],

Z ~ n ^{(T - AI)Z ~} n ^{(T - AI)H ~} n ^{(T - AI)H = {O},}

AEC AEC AEo-(T)

which implies ET( <!J)

{O}. By Theorem 1, there exists an idempotent operator R

£(H) such that R(H)

HT(F)

ET(F), since every normal operator is (E)-super-decomposable operator. By Theorem of [13] and Theorem 5.2 of [18], we have

HT(F) = ^E(F)H = n ^{(T - A)H.}

AfJ.F

This completes the proof.

Now we can prove the main result of this paper.

THEOREM

3. Let A be a semi-simple commutative complex Banach algebra with or without identity, and let a

A such that the multiplica- tion operator Ta on A has the weak 2-spectral decomposition property.

Then for any closed subset F of C, there exists an idempotent element rEA such that

Tr(A)

n (Ta - AI)A.

AfJ.F

Moreover, Tr(A) = ^Ar,,(F) = ^Er..(F) = ^{{x EA:}

a-

^{(F)} is}

an ideal of A, where

may be taken as the hull-kernel closure of the set {4> E 6(A) : x(4)) f:. O} in 6(A).

Proof. Assume that Ta has the weak 2-spectral decomposition prop- erty. At first, we claim that the Gelfand transform a is hull-kernel continuo\J.s on 6(A). Suppose that a is not continuous for the hull- kernel topology of 6(A). Then there exists closed W

e ^{such that}

a-I(W) is not a hull in 6(A). Let 4>

h(k(a-

(W))) \ a-

(W) and let A

4>(a) ft ^a-I(W). Since Ta has the weak 2-spectral decomposition property, there exists Y, Z

Lat(T

such that

A = Y + ^Z,u(Ta\Y)

e \

^{and u(TaIZ)}

e \ a-

^(W).

For each

Y there exists u

Y such that

= (Ta - A)U. Thus 4>(y) = ^{(4)(a) - A)4>(U)} = 0 and hence 4> = ^{0 on Y Let} .,p

a-I(W) and

.,p( a)

W. For each z

Z there exists v

Z such that (Ta - p.)v = ^{z. Thus .,p(z)} = (.,p(a) - p.).,p(v) = 0 and so "p = ^{0 on}

Z. It follows from 4>

h(k(a-

(W))) that 4> = 0 on Z. Thus 4> = 0 on A = Y + ^{Z. Since 4>}

^6(A), this contradicts our assumption that

a is hull-kernel continuous on 6(A). Let Al

A EI7 e ^{denote the}

unitization of a given commutative complex Banach algebra A without identity. It is clear that Al is semi-simple and a is also continuous

on 6(A

= 6(A) U{ 4>00}, where each 4>

6(A) is identified with

its canonical extension to Al and 4>00( a + AI)

A for all a E A and A

e. Without loss of generality, we may assume that A is semi- simple,

Banach algebra with identity. Given an arbitrary open cover {Ut, U

of e, choose a pair of open sets Wt, W

e such that

e \ ^UI

WI

WI

W

W

U

Then both a-I(W

and a-l(e \ W

are compact with respect to the hull-kernel topology of 6(A). By Corollary 3. 6. 10 of [17], there exists rEA such that

Since ;:z = r, we obtain r

= r, by the semi-simplity of A. Let

Tr E .L:(A) be given by Trx

rx for all x E A. It is easily checked

that TaTr

TrTa ^and T; = Tr. ^Let A

C \ U

Then clearly

Apply Theorem 3. 6. 15 of [17] to obtain e E A such that (a - A1)cr

r on 6(A), which implies (a - A1)er

r, i.e., (Ta - AI)STr

Tr where Sx

ex for all x

A. Hence (Ta - AI)ITr(A) is invertible and so O'(TaITr(A»

U

Proceeding in the same way, we obtain O'(Tal(I - Tr)(A»

and hence Ta is (E)-super-decoposable. If Y

A is Ta-divisible subspace, then we have

Y

n ^{(Ta - AI)Y ~} n ^{(Ta - AI)A ~ rad (A),}

~EC ~EC

where rad (A) denotes the radical of A. The semi-simplicity of A im- plies that

Y

n ^{(Ta - AI)A}

^{O},

~EC

and so E T(4)) ⁼ {O}. Hence by Theorem 1, Tr(A)

ETa (F)

ATa(F).

Since the spectral maximal space ATa(F)

Tr(A) is hyperinvariant, A Ta ^(F) is an ideal of A. On the other hand, it follows from Theorem 6. 2. 5 of [5] that

ATa(F)

{x EA:

a-I (F)}.

Finally, we claim that

ATa(F)

n (Ta - AI)A.

M/.F

The inclusion A Ta (F) ~ n ^{(Ta - AI)A} is clear from the elemen-

~EC\F

tary fact that (Ta - A)ATa(F)

AT,.{F) for all A

F. To prove the reverse inclusion, take any open neighborhood V of F. Then {V, C \ F} is an open covering of the complex plane C. As Ta is (E)-super- decomposable, there exists an idempotent RE LeX) commuting with Ta such that

O'(TaIR(A»

V and O'(Tal(I - R)(A»

C \ F.

Let

z:= n (Ta - -\1)A.

>.r;.F It

is easily checked that .

(1 - R)(Z) _~ n (Ta - -\l)A _~ rad(A),

>.ec

which implies that Z

R(A)

AT.. (V). Since XT(-) is known to preserve intersection, Z _~ n ^{AT.. (V)}

^{AT.. (F).} Hence we have

F~V

AT.. (F)

n (Ta - >.l)A.

>.r;.F

This completes the proof.

Note that the preceding proof shows that actually Ta is (E)-super- decompsable if and only if Ta is super-decompsable if and only if Ta is decompsable if and only if Ta has the weak 2-spectral decomposition property if and only if the Gelfand transform Cl is hull-kernel continuous on

THEOREM

4. Let A be a commutative complex Banach algebra with or without identity, and let a

A such that the Gelfand transform Cl is hull-kernel continuous on

Then for any closed F

C, there exists an idempotent element [r]

B

A/rad (A), the coset ofr

A, such that

T[r](B)

=

=

= n ^{(1[a] -}

>.r;.F

--.. ----1

= {

[x]

B : supp [x]

[a] (FH, where rad (A) denotes the radical of A.

Proof. It is clear that B

A/rad(A) is a semi-simple commutative Banach algebra. Let

A

A/rad (A) be the quotient map. Then by Theorem 23.5 of [4], the map

1r* : ~(A/rad

(A»

defined by

for

6(A/rad (A)) is a homeomorphism.

Hence 1;(;;)

~ is hull-kernel continuous on 6(A/rad (A)), since a ^is

hull-kernel contiuous on 6(A). It is easily seen that T[a] : A/rad (A) ---. A/rad (A),

given by T[a] [x]

lax] for all [x]

A/rad (A), is (E)-super-decomposable.

By Theorem 3, one has the results. This completes the proof.

An obvious combination of Theorem 3 and Theorem 2 of [6] leads to the following results.

COROLLARY

5. Let A be

semi-simple, regular commutative com- plex Banach algebra with or without identity, and let a E A. Then for any closed subset F of C, there exists an idempotent element rEA such that

Tr(A) = A T4 (F) = ET4(F) = n (Ta - '\I)A.

AfJF

The following Corollary applies, for instance, to any commutative C*-algebra, since every commutative C*-algebra is semi-simple and regular Banach algebra. We have the following.

COROLLARY

6. Let A be a commutative C*-algebra, and let a E A.

Then for any closed F

C, there exists an idempotent element rEA such that

Tr(A) = _AT4(F) = ET,.{F) = n (Ta - '\I)A.

AfJF

Proof. Every commutative C*-algebra is regular and the hull-kernel topology coincides with the Gelfand topology on 6(A). Thus Ta has the weak 2-spectral decomposition property. By Theorem 3, one has the results.

THEOREM

7. Let X be a complex Banach space, and let A denote a

semi-simple commutative Banach algebra with identity. Assume that

ep : A ---. £(X) is an algebra homomorphism. H Ta E £(A) has

the weak 2-spectral decomposition property, then the corresponding operator T

I)(a)

.c(X) is (E)-super-decomposable.

Proof. It is easily checked that the Gelfand transform a is hull-kernel continuous on .6(A). For every open covering {U, V} ofC, choose a pair of open sets G,

C such thatG

U,K

V and GUK

C. Then both a-I(C \ G) and a-I(C \ K) are disjoint, compact with respect to the Gelfand topology of the compact space .6(A). By Corollary 3. 6.10 of [17], we deduce that there exists bE A such that

~

Let R

I)(b)

.c(A). Clearly, T commutes with R. From

r on .6(A), we know that

=

by semi-simplicity of A and so R

= ^R.

We claim that u(TIR(X»

U. Let

C \ U. Then I(a - 'x1)1 >

holds on the hull a-leG). Thus we may apply Theorem 3.6. 15 of [17] to obtain dE A such that (a - 'x1)d

1 on a-leG). Since b

0 on a-I(C \ G), we have (a - 'x)bd

b on .6(A), which implies that .(a - 'x)bd

b. Let S

I)(d)

.c(X). Apply the homomorphism

to the equation (a - 'x1)bd

b, we have (T - 'xI)SR

SeT - 'xI)R

R.

Thus (T - 'xI)S = ^{SeT - 'xl)} = ^{I on R(X),} the range of R. Hence (T - 'xI)IR(X) is invertible, Le., ,x

p(TIR(X», and so u(TIR(X»

U. A similar argument can be used to show that u(TI(I - R)(X»

V.

Hence T is (E)-super-decomposable. This completes the proof.

COROLLARY

8. In the situation of Theorem 7, ifnAEcCT- 'x)X = {O}, then for any closed F

C, there exists an idempotent operator RE .c(X) such that

R(X)

XT(F)

ET(F)

n ^{(T - 'x)X.}

M/.F

Note that for a given commutative, semi-simple and regular Banach algebra A over C, the left regular representation

A

.c( A) given by 1)(a)

Ta for all a

A.

follows from [16] that Ta is (E)- super-decomposable. Moreover, every multiplication operator Ta on A satisfies

n ^{(Ta - 'x)A}

^rad(A),

AEC

the radical of A, so that n ^{(Ta - 'x)A}

^{{D}, since A} is semi-simple.

~EC

It

follows from Corollary 8 that for each closed F

C there exists an idempotent operator R

.c(A) such that

R(A)

AT.. (F)

ETJF)

n (Ta - 'x)A.

~~F

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Jae Chul Rho

Department of Mathematices Sogang University

Seoul 121-742, Korea Jong-Kwang Yoo

Department of Mathematices Chodang University

Muan 534-800, Korea