Range-Kernel orthogonality and elementary operators on certain Banach spaces

Abstract

The characterization of the points in Cp:1≤p<∞(ℋ){C}_{p{:}_{1\le p\lt \infty }}\left({\mathcal{ {\mathcal H} }}), the Von Neuman-Schatten p-classes, that are orthogonal to the range of elementary operators has been done for certain kinds of elementary operators. In this paper, we shall study this problem of characterization on an abstract reflexive, smooth and strictly convex Banach space for arbitrary operator. As an application, we consider other kinds of elementary operators defined on the spaces Cp:1≤p<∞(ℋ){C}_{p{:}_{1\le p\lt \infty }}\left({\mathcal{ {\mathcal H} }}), and finally, we give a counterexample to Mecheri’s result given in this context.

Keywords

• range-kernel orthogonality
• elementary operator
• schatten p-classes
• trace class operators
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