It is our purpose to obtain necessary and sufficient conditions on a sequence. IA} (n _> 0) of operators in B(X, Y) in order that itwill be a weak ora strong moment ...
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 vectorvalued measure in [0, 1] into B + such that
, defined on the Cfield 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 vectorvalued 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 semivariation 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 Cfield 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 vectorvalued 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 vectorvalued 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,
] JAml, .’, 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. AA. lowing consequence of Theorem 2. CORO,ZxaY 1. The sequence {A.} (n
AA.+.
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. 289305. 1. DUNFORD AND J. W. SCHWARTZ, Linear operators I, Interscience, New York, 1958. D. LEVIATAN, A generalized moment problem for selfadjoint operators, Israel J. Math., vol. 4, (1966), pp. 113118. I. J. SCHOENBER(, On finite rowed systems of linear inequalities in infinitely many variables, Trans. Amer. Math. Soc., vol. 34, (1932), pp. 594619. D. H. TUCKER, A note on the Riesz representation theorem, Proc. Amer. Math. Soc., vol. 14 (1963), pp. 354358. A representation theorem for a continuous linear transformation on a space of continuous functions, Proc. Amer. Math. Soc., vol. 16 (1965), pp. 946953.
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