Operators on Partial Inner Product Spaces: Towards a Spectral Analysis

TL;DR

This paper develops a spectral analysis framework for symmetric operators in partial inner product spaces, introducing a generalized resolvent and exploring eigenvalues related to the continuous spectrum.

Contribution

It extends spectral theory to operators on LHS, defining a generalized resolvent and analyzing continuous spectrum eigenvalues using the KLMN theorem.

Findings

Defined a generalized resolvent for operators in partial inner product spaces

Analyzed the spectral properties and eigenvalues related to the continuous spectrum

Provided examples including frame multipliers

Abstract

Given a LHS (Lattice of Hilbert spaces) $V_J$ and a symmetric operator $A$ in $V_J$, in the sense of partial inner product spaces, we define a generalized resolvent for $A$ and study the corresponding spectral properties. In particular, we examine, with help of the KLMN theorem, the question of generalized eigenvalues associated to points of the continuous (Hilbertian) spectrum. We give some examples, including so-called frame multipliers.