P-Operators on Hilbert Spaces

TL;DR

This paper extends the concept of P-matrices to P-operators on separable real Hilbert spaces, exploring their properties relative to different orthonormal bases and building on recent infinite-dimensional generalizations.

Contribution

It introduces the notion of P-operators on Hilbert spaces and examines their properties relative to various orthonormal bases, expanding the theory beyond finite-dimensional matrices.

Findings

P-operators are characterized on Hilbert spaces.

The relation between P-operators and orthonormal bases is analyzed.

The work generalizes finite-dimensional P-matrix concepts to infinite-dimensional settings.

Abstract

A real square matrix $A$ is called a P-matrix if all its principal minors are positive. Using the sign non-reversal property of matrices, the notion of P-matrix has been recently extended by Kannan and Sivakumar to infinite-dimensional Banach spaces relative to a given Schauder basis. Motivated by their work, we discuss P-operators on separable real Hilbert spaces. We also investigate P-operators relative to various orthonormal bases.