Adjoint Operators on Banach Spaces

TL;DR

This paper explores the existence of adjoint operators on separable Banach spaces, extending classical results and applying them to inequalities, spectral theory, and operator classes, with implications for path integral construction.

Contribution

It introduces new results on adjoint existence for operators on Banach spaces, extending classical theorems and defining a new class of Banach spaces.

Findings

Extended Poincaré inequality to Banach spaces

Generalized spectral theorem for $C_0$-generators

Extended Schatten-class to all separable Banach spaces

Abstract

In this paper, we report on new results related to the existence of an adjoint for operators on separable Banach spaces and discuss a few interesting applications. (Some results are new even for Hilbert spaces.) Our first two applications provide an extension of the Poincar\'{e} inequality and the Stone-von Neumann version of the spectral theorem for a large class of $C_0$-generators of contraction semigroups on separable Banach spaces. Our third application provides a natural extension of the Schatten-class of operators to all separable Banach spaces. As a part of this program, we introduce a new class of separable Banach spaces. As a side benefit, these spaces also provide a natural framework for the (rigorous) construction of the path integral as envisioned by Feynman.