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
Summary
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.
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