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U-ideals of factorable operators

Kamil John — 1999

Czechoslovak Mathematical Journal

We suggest a method of renorming of spaces of operators which are suitably approximable by sequences of operators from a given class. Further we generalize J. Johnsons’s construction of ideals of compact operators in the space of bounded operators and observe e.g. that under our renormings compact operators are u -ideals in the: space of 2-absolutely summing operators or in the space of operators factorable through a Hilbert space.

w * -basic sequences and reflexivity of Banach spaces

Kamil John — 2005

Czechoslovak Mathematical Journal

We observe that a separable Banach space X is reflexive iff each of its quotients with Schauder basis is reflexive. Similarly if ( X , Y ) is not reflexive for reflexive X and Y then ( X 1 , Y ) is is not reflexive for some X 1 X , X 1 having a basis.

Projections from L ( X , Y ) onto K ( X , Y )

Kamil John — 2000

Commentationes Mathematicae Universitatis Carolinae

Generalization of certain results in [Sap] and simplification of the proofs are given. We observe e.g.: Let X and Y be Banach spaces such that X is weakly compactly generated Asplund space and X * has the approximation property (respectively Y is weakly compactly generated Asplund space and Y * has the approximation property). Suppose that L ( X , Y ) K ( X , Y ) and let 1 < λ < 2 . Then X (respectively Y ) can be equivalently renormed so that any projection P of L ( X , Y ) onto K ( X , Y ) has the sup-norm greater or equal to λ .

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