Partial inner product spaces: Some categorical aspects

TL;DR

This paper explores the categorical structure of partial inner product spaces (PIP spaces), explicitly constructing sheaves and cosheaves of operators to better understand their theoretical framework.

Contribution

It introduces a categorical perspective to PIP spaces and constructs sheaves and cosheaves of operators, providing new tools for their analysis.

Findings

Categorical formulation of PIP spaces clarified

Construction of sheaves and cosheaves of operators

Enhanced understanding of PIP spaces in practical contexts

Abstract

We make explicit in terms of categories a number of statements from the theory of partial inner product spaces (PIP spaces) and operators on them. In particular, we construct sheaves and cosheaves of operators on certain PIP spaces of practical interest.