Construction of strong derivable maps via functional calculus of unbounded spectral operators in Banach spaces

TL;DR

This paper establishes conditions for the existence and calculation of strong derivable maps via functional calculus of unbounded spectral operators in Banach spaces, extending classical results to locally convex spaces.

Contribution

It generalizes classical spectral operator results to unbounded operators in Banach and locally convex spaces, providing new methods to compute derivatives of functional calculus maps.

Findings

Derived conditions for strong derivability of maps in spectral calculus.

Calculated derivatives using spectral operator functional calculus.

Extended classical Banach space results to locally convex spaces.

Abstract

We provide sufficient conditions for the existence of a strong derivable map and calculate its derivative by employing a result in our previous work on strong derivability of maps arising by functional calculus of an unbounded scalar type spectral operator $R$ in a Banach space and the generalization to complete locally convex spaces of a classical result valid in the Banach space context. We apply this result to obtain a sequence of integrals converging to an integral of a complete locally convex space extension of a map arising by functional calculus of $R$.