Interpolative construction for operator ideals

Abstract

The problem from which this article originated is the following: given an operator $T:E→ F$ between Banach spaces belonging simultaneously to two operator ideals, $\mathcal A$ and $\mathcal B$ say, when is it possible to find a decomposition $T = A· B$, where $A∈ \mathcal A$ and $B∈\mathcal B$, or at least $A∈ \dot{\mathcal{A}}$ and $B∈ \ddot{\mathcal B}$, with $\dot{\mathcal{A}}$ and $\ddot{\mathcal B}$ being associated with $\mathcal A$ and $\mathcal B$ in a specific sense? It was shown by S. Heinrich [2] that such a decomposition is always possible, with $\mathcal A=\dot{\mathcal{A}}$ and $\mathcal B=\ddot{\mathcal B}$,if $\mathcal A$ and $\mathcal B$ are uniformly closed, $\mathcal A$ is surjective, and $\mathcal B$ is injective.Heinrich’s arguments are based on a simple interpolation technique which appears to be strongy related to certain general constructions with operator ideals that were successfully applied in a seemingly different context in recent years (ref.[8],[5],and [4]-[7], [1]). We intend to investigate the fundamentals of such constructions and their interpolation-theoretic background in this paper, with emphasis on the impact to the factorization problem.Applications will be given for ideals generated by s-number sequences and to type p and cotype q operators.