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  • Cited by 1
  • Print publication year: 1989
  • Online publication date: August 2010

11 - On some operator ideals defined by approximation numbers

Summary

Abstract. We prove a representation theorem in terms of finite rank operators for operators belonging to. Some information on the tensor product of operators belonging to these ideals is also obtained.

INTRODUCTION.

The n-th approximation number an (T) of a bounded linear operator T∈ (E, F) acting between the Banach spaces E and F, is defined as

For the ideal is formed by all operators T betwen Banach space, with a finite quasi-norm

Weyl ideals are defined in a similar way, by substituting approximation numbers for Weyl numbers (xn (T)). Ideals have been studied by the authors in. Since xn (T) ≤ an (T) for al 1 n ∈ N (see), as a direct consequence of, Thm. 3, we have

Theorem 1. Let 0 < n ∞. Then there is a constant M = M such that for any complex Banach space E and any operator T ∈∞, ∞ (E, E) the following holds

Here (λn (T)) denotes the sequence of all eigenvalues of the compact operator T counted accoding to their algebraic multiplicities and ordered such that |λ1 (T)| ≥ |λ2(T)| ≥ … ≥ 0.

In this note we continue the study of -ideals. We derive a representation theorem for the elements of in terms of finite rank operators. This result is on the same lines as we established in for the case of the ideals (0 < q < ∞). We also obtain some information on the tensor pordect of operator belonging to the scale of the ideals.