Abstract

The goal of this paper is to develop the theory of Schur complementation in the context of operators acting on anti-dual pairs. As a byproduct, we obtain a natural generalization of the parallel sum and parallel difference, as well as the Lebesgue-type decomposition. To demonstrate how this operator approach works in application, we derive the corresponding results for operators acting on rigged Hilbert spaces, and for representable functionals of ⁎-algebras.

Highlights

  • Since the first appearance of the name of the “Schur complement” in [10], the theory of partitioned matrices is an active field of research in linear algebra and functional analysis

  • The goal of this paper is to develop the theory of Schur complementation in the context of operators acting on anti-dual pairs

  • Following the method of Pekarev and Šmul’jan, we introduce the notions of parallel sum and difference as an immediate application

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Summary

Introduction

Since the first appearance of the name of the “Schur complement” in [10], the theory of partitioned matrices (or block operators) is an active field of research in linear algebra and functional analysis.

B D is a positive semidefinite
Operators on anti-dual pairs
Generalized Schur complement
Operators on rigged Hilbert spaces
Representable functionals
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