(iii) b*A, fo,(t)(dt)b* n - Project Euclid

ON MOMENT SEQUENCES OF OPERATORS BY

DANY LEVIATAN 1. Introduction

_

_

Let X, Y be Banach spaces over the complex field and denote by B B (X, Y) the space of continuous linear operators on X into Y. Recently Tucker [6] has introduced a weak extension Y+ of the Banach space Y and has proved that B+ B(X, Y+). The weak extension Y+ is by construction a denotes the closure of B + in B** (X, Y) subspace of Y**, consequently if ’’$ B(X, Y**). topologized in the natural way we obtain B DEFINITION 1. Given a sequence/(t)}(n _> 0) C[O, 1], the sequence IA,I B(X, Y) is called a weak moment sequence with respect to

’

if there exists a vector-valued measure in [0, 1] into B + such that

, defined on the C-field of Borel sets

_

-

(i) ( )b* is in rca [0, 1] for each b* eB*(X, Y); (ii) the mapping b* ( )b* is continuous with the B(X, Y) and C[0, 1] topologies of B* (X, Y) and rca [0, 1] respectively; b* e B*(X, Y); (iii) b*A, fo,(t)(dt)b* n O, 1, 2, sup I] aCz(E,) < (iv) I I1[0, where the supremum is taken over all finite collections of disj oint Borel sets in [0, 1] and all finite sets of scalars a with ai[ 1.

,

DEFINITION 2. Given a sequence [(t)} C[0, 1], the sequence B(X, Y) is called a strong moment sequence with respect to if there exists a vector-valued measure z, defined on the -field of Borel sets in [0, 1] into B(X, Y) such that

[A}

(i) b*( is in rca [0, 1], b* eB* (X, Y); (ii) A, f ,(t)(dt) n 0,1,2,...; (iii) ll[O, 1]
0) of operators in B(X, Y) in order that it will be a weak or a strong moment sequence with respect to ICe(t)} (n >_ 0) in various cases of sequences ICe(t)}. We shall be interested, especially, in the case where n >_ 0, where the sequence k} (n _> 0) satisfies (t)

x,

(1.1) O 0 for every 1(i) (if m a. 1) and assume that (, 0 _< il < < ira. For a given sequence of operators IAn} (n _> 0) define (2.1) D’A= --0(-1)J(i,’",i+j- 1, i+j+l,...,i+k)A.j,

...,

k>O,

DA A and

(0,m+ 1,... ,...n) D’-mA,,,, 0 0 for0_< t_< 1 and 0_< m _ O) is a fundamental set in C[O, 1], that is, {),(t)} (n _> O) spans C[O, 1] in the maximum norm. Then the sequence {A,} (n >_ O) of operators in B(X, Y) is a wea moment sequence with respect to the sequence

(3.1)

sup

...,

a, with where the supremum is taken over all the finite set of scalars ao, 1 and all n > O. Moreover the semi-variation t [[[0, 1] M. a,, Proof. Suppose, first, that (3.1) holds and define the operator

M we have Hence T M. By the representation theorem of operators from C[0, 1] to T B(X, Y) (see [2, Theorem VI. 7.2] or [1, Theorem 3.1]) there exists a vector valued measure from the C-field of Borel sets in [0, 1] to B** (X, Y) satisfy-

,

ing conditions (i) and (ii) of Definition 1 such that

(3.4)

f C[O, 1], b* e B*(Z, Y)

fof(t)t(dt)b*,

b*T(f)

and

IIT

(3.5) By (3.4) D*T(rh,) b *A, and by (3.5)

li[o, 1].

frh(t)(dt)b*,

n

O, 1, 2,

..

b* B* (X, Y)

M By the construction of in the proof of Theorem VI. 7.2 [2] and using the arguments similar to [5] and [6] one can easily prove that for each closed set F % [0, 1], (F) e B+ and thus it is readily seen that for every Borel set E [0, 1]. (E) e Thus we proved that IA} (n >_ 0) is a weak moment sequence with respect to {(t)} (n >_ 0). Conversely, suppose that {Am} (n >_ 0) is a weak moment sequence with respect to { (t)} (n _> 0). The vector-valued measure existing by Definition

I1[o, 1]

1 defines an operator

-

,(E,)II

sup

T C[0, 1]-+ B(X, Y) by the equation (3.4) (see [2, Theorem VI. 7.2] ). The operator T is bounded and satisfies (3.5). Now b*(=0amkm) b*T(.,=oa,),n,(t)) for every b* e B* (X, Y) hence

Enm=O Olm)knm

T( :0 a:Xnm(t)

whence

-o

o>, < T (3.6) For every fite set of scalars ao,

supo_ 0. Suppose that the sequence {A} (n 0) is a 0). The weak moment sequence with respect to the sequence {t’} (n vector-valued measure existing by Definition 1, defines (see [2] Theorem B(X, Y) by the equation (3.4). Define VI.7.2) an operator T’C[O, 1] sequences {}, {n} (n _> 0) by

,

(3.11)

0

T(1),

n

A_(n

1), 0

0,

n =),,_(n_ .1),

then the sequence {fi} (n _> 0) is a weak moment sequence with respect to the sequence {ti’} (n 0). The sequence {,,} (n 0) satisfies (1.1) with0 0

and for this case we have already proved Theorem 2. It is readily seen that for n>m>l

[m,

] JAm-l, .’-, A_I], hence by (3.8) for the sequence }, } (n _> 0), (3.12)

"",

sup sup

, ...,

Convemely, if wih one exception, 0 i n rbimry bounded opera,or. By (2.3) for h nd (g.9) for he equence ) ( 2 0) we ge

[go,

1)x

hence by (3.12)

X, Now, if[a[

g]

o -x (-- 1)-+ [g, g], X. ]l[go, ..., X_x[A_x, , A_x]]. ][ o + [ -x (-- l)1for0

m n,

,,,-o am+" hm+

-’F"

, A._ 11,

M-,[Am,

hence by (3.8) and (3.13)

_< i Ao I

-

+ 2M.

Thus we have proved that

(3.14)

a,with where the supremum is taken over all the finite sets of scalars a0, Jan 1 and all n _> 0. As the sequence {} (n _> 0) satisfies (1.1) with ,0 0 we obtain by (3.14) and Theorem 2, which we have proved for this case, that the sequence I} (n _> 0) is a weak moment sequence with respect to {t.l (n >_ 0). This implies the desired conclusion. Q.E.D. For the sequence n} (n >_ O) we have k,,,,,

()A"-mA,,,,

where AA. A. and AA. A-A. lowing consequence of Theorem 2. CORO,ZxaY 1. The sequence {A.} (n

A-A.+.

0

_
_ 0) of operators in B(X, Y) s a weak

_

ON MOMENT SEQUENCES OF OPERATORS

-

moment sequence with respect to the sequence {tn} (n sup

,=oa()A"-A

255

_> 0) if and only if M
_ 0) is fundamental in [0, 1] and that Y is reflexive. Then the sequence A n >_ O) of operators in B(X, Y) is a strong moment sequence with respect to {,(t) (n _> 0) if and only if (3.1) holds. Moreover 1] i![0, 1] M. Proof. If Y is reflexive, then the measure obtained by Theorem 1 takes values in B (X, Y) and the proof of our theorem is similar to that of [2] Theorem I.7.3, Q.E.D. Similarly we obtain

THEOREM 4. Let {),,} (n >_ 0) satisfy (1.1) and suppose that Y is reflexive. Then the sequence {Am} (n >_ 0) of operators in B(X, Y) is a strong moment sequence with respect to {tx"} (n _> 0) if and only if (3.8) holds. Moreover, if o O, then I!, I1[O, 11 i. Theorem 3 is a generalization of [3, Theorem 1], and Theorem 4 is a generalization of [3, Consequence 2 and Theorem 2]. 1. 2.

3. 4.

5. 6.

REFERENCES R. G. BARTLE, N. DUNFORD AND J. T. SCHWARTZ, Weak compactness and vector measures, Canadian J. Math., vol. 7, (1955), pp. 289-305. 1. DUNFORD AND J. W. SCHWARTZ, Linear operators I, Interscience, New York, 1958. D. LEVIATAN, A generalized moment problem for self-adjoint operators, Israel J. Math., vol. 4, (1966), pp. 113-118. I. J. SCHOENBER(, On finite rowed systems of linear inequalities in infinitely many variables, Trans. Amer. Math. Soc., vol. 34, (1932), pp. 594-619. D. H. TUCKER, A note on the Riesz representation theorem, Proc. Amer. Math. Soc., vol. 14 (1963), pp. 354-358. A representation theorem for a continuous linear transformation on a space of continuous functions, Proc. Amer. Math. Soc., vol. 16 (1965), pp. 946-953.

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