Interpolative construction for operator ideals
Abstract
The problem from which this article originated is the following: given an operator 
between Banach spaces belonging simultaneously to two operator ideals, 
and 
say, when is it possible to find a decomposition 
, where 
and 
, or at least 
and 
, with 
and 
being associated with and in a specific sense? It was shown by S. Heinrich [2] that such a decomposition is always possible, with 
and 
,if and are uniformly closed, is surjective, and 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.
between Banach spaces belonging simultaneously to two operator ideals, and say, when is it possible to find a decomposition , where and , or at least and , with and being associated with and in a specific sense? It was shown by S. Heinrich [2] that such a decomposition is always possible, with and ,if and are uniformly closed, is surjective, and 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.DOI Code:
10.1285/i15900932v8n1p45
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