Adjoint for Operators in Banach Spaces

TL;DR

This paper extends fundamental operator theorems to Banach spaces using a result on Gaussian measures, introduces an adjoint for operators, and develops new approximation and metric tools for operator analysis.

Contribution

It introduces an adjoint for operators in Banach spaces, extends classical theorems, and develops new approximation methods and metrics for operators.

Findings

Constructed an adjoint for operators on separable Banach spaces.

Extended von Neumann and Lax theorems to Banach spaces.

Provided a new metric for closed densely defined operators.

Abstract

In this paper we show that a result of Gross and Kuelbs, used to study Gaussian measures on Banach spaces, makes it possible to construct an adjoint for operators on separable Banach spaces. This result is used to extend well known theorems of von Neumann and Lax. We also partially solve an open problem on the existence of a Markushevich basis with unit norm and prove that all closed densely defined linear operators on a separable Banach space can be approximated by bounded operators. This last result extends a theorem of Kaufman for Hilbert spaces and allows us to define a new metric for closed densely defined linear operators on Banach spaces. As an application, we obtain a generalization of the Yosida approximator for semigroups of operators.