Conditions of fixed sign for $n\times n$ operator matrices

TL;DR

This paper establishes criteria for positive definiteness and nonnegativity of self-adjoint operator matrices in Hilbert spaces, with applications to conditions for functional extremum of multiple variables.

Contribution

It introduces new criteria for positive definiteness and nonnegativity of operator matrices, extending to multiple Hilbert spaces and applying to extremum problems.

Findings

Derived positive definiteness criterion for operator matrices.

Established nonnegativity conditions under additional assumptions.

Applied criteria to analyze functional extremum of multiple variables.

Abstract

A positive definiteness criterion and, under the additional conditions, a nonnegativity criterion for a self-adjoint continuous operator matrix, acting in product of an arbitrary number of real separable Hilbert spaces, are obtained. As application, both sufficient and necessary analytical conditions of functional extremum of several Hilbert variables are considered.