Real and Complex Operator Ideals

The powerful concept of an operator ideal on the class of all Banach spaces makes sense in the real and in the complex case. In both settings we may, for example, consider compact, nuclear, or 2–summing operators, where the definitions are adapted to each other in a natural way. This paper deals with the question whether or not that fact is based on a general philosophy. Does there exists a one– to–one correspondence between “real properties” and “complex properties” defining an operator ideal? In other words, does there exist for every real operator ideal a uniquely determined corresponding complex ideal and vice versa? Unfortunately, we are not abel to give a final answer. Nevertheless, some preliminary results are obtained. In particular, we construct for every real operator ideal a corresponding complex operator ideal and for every complex operator ideal a corresponding real one. However, we conjecture that there exists a complex operator ideal which can not be obtained from a real one by this construction. The following approach is based on the observation that every complex Banach space can be viewed as a real Banach space with an isometry acting on it like the scalar multiplication by the imaginary unit i. 1. Preliminaries Let X always denote a Banach space over the field of real numbers. The letters A, B, T and S refer to linear operators between real Banach spaces. The identity map of X is denoted by IX . For x ∈ X the canonical injections X → X ⊕X : x 7→ (x, o) and X → X ⊕X : x 7→ (o, x) are denoted by J 1 and J X 2 , respectively. For x1, x2 ∈ X the canonical surjections X ⊕X → X : (x1, x2) 7→ x1 and X ⊕X → X : (x1, x2) 7→ x2 are denoted by Q1 and Q X 2 , respectively. Let L always denote the ideal of all (real or complex) bounded linear operators. 1991 Mathematics Subject Classification. 46B20, 47D50. This article originated from the author’s masters thesis at the University of Jena written under the supervision of A. Pietsch