P-representable operators in Banach spaces

Roshdi Khalil

doi:10.1155/s0161171286000819

Abstract

LetEandFbe Banach spaces. An operatorT∈L(E,F)is calledp-representable if there exists a finite measureμon the unit ball,B(E*), ofE*and a functiong∈Lq(μ,F),1p+1q=1, such thatTx=∫B(E*)〈x,x*〉g(x*)dμ(x*)for allx∈E. The object of this paper is to investigate the class of allp-representable operators. In particular, it is shown thatp-representable operators form a Banach ideal which is stable under injective tensor product. A characterization via factorization throughLp-spaces is given.

Highlights

For all x E The object of this paper is to investigate the class of all p-representable operators
B(E*) the unit ball of E*, the dual of E The completion of the injective tensor product of E and F is denoted by E F Integral operators in L(E,F)
Were first defined by Grothendieck, 12], as those operators which can be identified with elements in (E F)*

Summary

If p

L’(’,,E) is the space of es3ecia!]y bounded functions on L with values ir, E. An operator T L(E,F) is called p-representable operator if there exists a finite measure defined on the Borel sets of B(E*) and a function iBfE,,, g’B(E*) --F such that ilg(x*)ll qd , and Tx x,x* g(x*)d(x*). It follows from the definition that every p-representable operator is Pietsch-p-. Let be the set of all p-representable operators from E into F. -x,x* g(x*)d..(x*), for some finite measure on B(E*) and some g E Lq(B(E *),u, F) By definition of approximable operators, Pietsch 16] T is approximable. Rp(H,F), To show TB gn let be a sequence of simple functions converging to g in Lq(B(E*), F) and Tn be the associated operators in Rp (E F).

However Tn B

We let

Since the map