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Linear inessential operators and generalized inverses

Bruce A. Barnes — 2009

Commentationes Mathematicae Universitatis Carolinae

The space of inessential bounded linear operators from one Banach space X into another Y is introduced. This space, I ( X , Y ) , is a subspace of B ( X , Y ) which generalizes Kleinecke’s ideal of inessential operators. For certain subspaces W of I ( X , Y ) , it is shown that when T ∈ B ( X , Y ) has a generalized inverse modulo W , then there exists a projection P ∈ B ( X ) such that T ( I - P ) has a generalized inverse and T P ∈ W .

Diameter-preserving maps on various classes of function spaces

Bruce A. Barnes;  Ashoke K. Roy — 2002

Studia Mathematica

Under some mild assumptions, non-linear diameter-preserving bijections between (vector-valued) function spaces are characterized with the help of a well-known theorem of Ulam and Mazur. A necessary and sufficient condition for the existence of a diameter-preserving bijection between function spaces in the complex scalar case is derived, and a complete description of such maps is given in several important cases.

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