Stability of some operator ideals

Abstract

Abstract : An operator ideal O is said to be tensor stable with respect to a tensor norm Il lla if for all T1 € O(X1, Y1) and T2 € O(X2, Y2), T1 8T2 € O(X1 & X2, Y1 Y2). The earlist examples of tensor stable ideals with respect to || ||e are those of compact operators and absolutely p-summing operators. These results are due to Vala [14] and Holub [5], respectively. In 1988, Carl, Defant and Ramanujan, [2], identified E- or T-stability of some operator ideals including those discussed in our thesis. The primary goal of this thesis is to give simple proofs for e-stability of the ideals of p-summing operators, p-integral operators and p-nuclear operators using factorization techniques. The thesis also provides a proof for the e-stability of the injective hull of an e-stable operator ideal as well as for the T-stability of the ideal of comnact operators. The thesis consists of three chapters. In Chapter I, basic definitions and results about tensor product spaces are given. In particular, it is shown that C(I) & X = C(I,X) and L1(I) X= L1(I,X). In these theorems the unit interval equipped with the Lebesgue measure can be replaced with any compact Hausdorff space and any measure space, respectively. In Chapter II, the most important examples of operator ideals are considered. In fact, factorization theorems about the p-summing operators, the p-nuclear operators and the p-integral operators are presented in sections 2, 3, and 4. Finally, Chapter III presents our work on stability of operator ideals. These results are not new but the methods for the proof are new and simpler than those given by Carl, Defant and Ramanujan, [2].