AN OPERATOR SUMMABILITY OF SEQUENCESIN BANACH SPACES

作者：Karn Anil Kumar^*; Sinha Deba Prasad

来源：Glasgow Mathematical Journal, 2014, 56(2): 427-437.

DOI：10.1017/S0017089513000360

Let 1 %26lt;= p %26lt; infinity. A sequence %26lt; x(n)%26gt; in a Banach space X is defined to be p-operator summable if for each %26lt; f(n)%26gt;. l(p)(w*) ( X*) we have %26lt;%26lt; f(n)(x(k))%26gt;(k)%26gt; (n) is an element of l(p)(s) (lp()). Every norm p-summable sequence in a Banach space is operator p-summable whereas in its turn every operator p-summable sequence is weakly p-summable. An operator T is an element of B(X, Y) is said to be p-limited if for every %26lt; x(n)%26gt; is an element of l (w)(p) (X), %26lt; Tx(n)%26gt; is operator p-summable. The set of all p-limited operators forms a normed operator ideal. It is shown that every weakly p-summable sequence in X is operator p-summable if and only if every operator T is an element of B(X, l(p)) is p-absolutely summing. On the other hand, every operator p-summable sequence in X is norm p-summable if and only if every p-limited operator in B(l(p) , X) is absolutely p-summing. Moreover, this is the case if and only if X is a subspace of L-p(mu) for some Borel measure mu.

• 出版日期2014-5