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Computability of compact operators on computable Banach spaces with bases
Author(s) -
Brattka Vasco,
Dillhage Ruth
Publication year - 2007
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200710004
Subject(s) - mathematics , computable analysis , unbounded operator , pure mathematics , schauder basis , hilbert space , computable number , compact operator on hilbert space , banach space , nuclear operator , approximation property , discrete mathematics , finite rank operator , bounded function , eberlein–šmulian theorem , computability , compact operator , algebra over a field , lp space , mathematical analysis , computer science , extension (predicate logic) , programming language
We develop some parts of the theory of compact operators from the point of view of computable analysis. While computable compact operators on Hilbert spaces are easy to understand, it turns out that these operators on Banach spaces are harder to handle. Classically, the theory of compact operators on Banach spaces is developed with the help of the non‐constructive tool of sequential compactness. We demonstrate that a substantial amount of this theory can be developed computably on Banach spaces with computable Schauder bases that are well‐behaved. The conditions imposed on the bases are such that they generalize the Hilbert space case. In particular, we prove that the space of compact operators on Banach spaces with monotone, computably shrinking, and computable bases is a computable Banach space itself and operations such as composition with bounded linear operators from left are computable. Moreover, we provide a computable version of the Theorem of Schauder on adjoints in this framework and we discuss a non‐uniform result on composition with bounded linear operators from right. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)