Equivalent Banach Operator Ideal Norms 1
نویسنده
چکیده
Let X,Y be Banach spaces and consider the w′-topology (the dual weak operator topology) on the space (L(X,Y ), ‖.‖) of bounded linear operators from X into X with the uniform operator norm. L ′ (X,Y ) is the space of all T ∈ L(X,Y ) for which there exists a sequence of compact linear operators (Tn) ⊂ K(X,Y ) such that T = w′ − limnTn. Financial support from the National Council for Science and Technology (NCST) is greatly acknowledged. 20 Musundi Sammy, Shem Aywa and Jan Fourie Two equivalent norms, ‖|T‖| := inf{ sup n ‖Tn‖ : Tn ∈ K(X,Y ), Tn w′ → T} and ‖T‖u := inf{ sup n {max{‖Tn‖, ‖T−2Tn‖}} : ‖ : Tn ∈ K(X,Y ), Tn w′ → T} on L ′ (X,Y ), are considered. We show that (L ′ , |‖.‖|) and (Lw , ‖.‖u) are Banach operator ideals. Mathematics Subject Classification: 47B10; 46B10; 46A25
منابع مشابه
ar X iv : m at h / 02 02 07 3 v 1 [ m at h . FA ] 8 F eb 2 00 2 UNIFORMLY CONVEX OPERATORS AND MARTINGALE TYPE
The concept of uniform convexity of a Banach space was generalized to linear operators between Banach spaces and studied by Beauzamy [1]. Under this generalization, a Banach space X is uniformly convex if and only if its identity map IX is. Pisier showed that uniformly convex Banach spaces have martingale type p for some p > 1. We show that this fact is in general not true for linear operators....
متن کاملEquivalence of Norms on Operator Space Tensor Products of C∗-algebras
The Haagerup norm ‖ · ‖h on the tensor product A ⊗ B of two C∗-algebras A and B is shown to be Banach space equivalent to either the Banach space projective norm ‖ · ‖γ or the operator space projective norm ‖ · ‖∧ if and only if either A or B is finite dimensional or A and B are infinite dimensional and subhomogeneous. The Banach space projective norm and the operator space projective norm are ...
متن کاملm at h . FA ] 1 2 A pr 1 99 4 IDEAL NORMS AND TRIGONOMETRIC ORTHONORMAL SYSTEMS
Abstract. In this article, we characterize the UMD–property of a Banach space X by ideal norms associated with trigonometric orthonormal systems. The asymptotic behavior of that numerical parameters can be used to decide whether or not X is a UMD–space. Moreover, in the negative case, we obtain a measure that shows how far X is from being a UMD–space. The main result is, that all described para...
متن کاملC0-semigroup and Operator Ideals
Let T (t), 0 ≤ t < ∞, be a one parameter c0-semigroup of bounded linear operators on a Banach space X with infinitesimal generator A and R(λ, A) be the resolvent operator of A. The Hille-Yosida Theorem for c0-semigroups asserts that the resolvent operator of the infinitesimal generator A satisfies ‖R(λ, A)‖ ≤ M λ−ω for some constants M > 0 and λ ∈ R (the set of real numbers), λ > ω. The object ...
متن کاملIdeal Properties of Regular Operators between Banach Lati ’ Ices
Suppose E and F are Banach lattices such that E* and F have order-continuous norms. In [4] Dodds and Fremlin (cf. also [1]) showed that if T: E F is a positive compact operator and 0 < S < T then S is also compact. Aliprantis and Burkinshaw [1] showed by examples that the hypotheses on E and F are necessary. In [2] they asked whether a similar result is true for Dunford-Pettis operators, under ...
متن کامل