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We give an explicit description of a tensor norm equivalent on to the associated tensor norm to the ideal of -absolutely summing operators. As a consequence, we describe a tensor norm on the class of Banach spaces which is equivalent to the left projective tensor norm associated to .
We study the space of p-compact operators, , using the theory of tensor norms and operator ideals. We prove that is associated to , the left injective associate of the Chevet-Saphar tensor norm (which is equal to ). This allows us to relate the theory of p-summing operators to that of p-compact operators. Using the results known for the former class and appropriate hypotheses on E and F we prove that is equal to for a wide range of values of p and q, and show that our results are sharp....
A Banach space X is said to have the weak λ-bounded approximation property if for every separable reflexive Banach space Y and for every compact operator T : X → Y, there exists a net (Sα) of finite-rank operators on X such that supα ||TSα|| ≤ λ||T|| and Sα → IX uniformly on compact subsets of X.We prove the following theorem. Let X** or Y* have the Radon-Nikodym property; if X has the weak λ-bounded approximation property, then for every bounded linear operator T: X → Y, there exists a net (Sα)...
Let be an operator ideal on LCS’s. A continuous seminorm p of a LCS X is said to be - continuous if , where is the completion of the normed space and is the canonical map. p is said to be a Groth()- seminorm if there is a continuous seminorm q of X such that p ≤ q and the canonical map belongs to . It is well known that when is the ideal of absolutely summing (resp. precompact, weakly compact) operators, a LCS X is a nuclear (resp. Schwartz, infra-Schwartz) space if and only if every continuous...
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