Functional calculus extensions on dual spaces

TL;DR

This paper demonstrates that Banach spaces with preduals allow operators with continuous functional calculus to extend to bounded Borel calculus, unifying finitely spectral and prespectral operators under these conditions.

Contribution

It establishes a link between continuous and bounded Borel functional calculus for operators on Banach spaces with preduals, and explores conditions for extending calculus types.

Findings

Operators with continuous calculus on such spaces admit bounded Borel calculus.

Finitely spectral and prespectral operators coincide on these spaces.

Provides conditions for operators with absolutely continuous calculus to have bounded Borel calculus.

Abstract

In this note, we show that if a Banach space X has a predual, then every bounded linear operator on X with a continuous functional calculus admits a bounded Borel functional calculus. A consequence of this is that on such a Banach space, the classes of finitely spectral and prespectral operators coincide. We also apply this result to give some sufficient conditions for an operator with an absolutely continuous functional calculus to admit a bounded Borel one.