2-Absolutely Summing Operators on the Space C(T,X)

Abstract

We give for some Banach spaces X and Y examples of linear and continuous operators U : C ( T , X ) + Y , such that U' cp E As,(X, Y ) , for each cp E C ( T ) and U # : C ( T ) + As,(X,Y) is a 2-absolutely summing operator with respect to the 2-absolute norm on As,(X, Y ) , but U is not 2-absolutely summing. o 1999 Aca deniic Press For X a Banach space, 1 I r I a, and ( x , , )~ c X we write wr(xn I n E N ) = sup{(C:=lIx*(x,,)I ) Ix* E X*, Ilx*ll I 11, if r < a and wx(xn I n E N ) = ~~p ,~~~l l x~l l , if r = m, w,(X> = { ( x ,~>~~~ cx I wr(x,, In E N )

Examples of summing, integral and nuclear operators on the space C([0,1],X) with values in Popa1 2007Journal of Mathematical Analysis and ApplicationsSelf Cite7000View full textAdd to dashboardCite

“…We remark that in [10, Theorem 2.1] is proved (a), for p-summing operators and in [15,Theorem 1], for (r, p)-summing operators, but as is easily seen from the proof, by symmetry, we get that (b) is also true.…”

“…In the next example: (b)(v) shows that for (2, 1)-summing operators, the converse of Theorem 3(a) is not true in general, which, as we know, is the first example of this kind; (b)(iv) and (v) are new examples different from those in [10] and [14] which show that the converse of Theorem 3(a) is not true in general; also we show that in Example 3(b)(iii), in this particular situation, the converse is true.…”

“…We begin with an example which appear in this form in [18, Theorems 2.1 and 3.1] and in a more general context in [14,Proposition 2]. Example 1.…”

“…In [10] and [14] examples which prove that the converse in Theorem 3(a) is not in general true, are given.…”

“…This will reveal, among others, what is hidden in these general theorems-in particular, whether the converses of the above general results are true or not. Further, in addition to [10] and [14], we consider whether the converse of Theorem 3(b) is true.…”