On Componental Operators in Hilbert Space

TL;DR

This paper introduces the concept of componental operators in product Hilbert spaces, analyzing their properties and fixed point sets, and extends fixed point theorems to these operators to improve iterative methods for sparse linear systems.

Contribution

It develops a framework for analyzing componental operators' properties and fixed points, extending fixed point theorems to these operators for better iterative methods.

Findings

Relationships between componental and full operators are established.

A variant of the Banach fixed point theorem for componental contractions is provided.

The framework facilitates accelerated convergence in sparse linear system solutions.

Abstract

We consider a Hilbert space that is a product of a finite number of Hilbert spaces and operators that are represented by "componental operators" acting on the Hilbert spaces that form the product space. We attribute operatorial properties to the componental operators rather than to the full operators. The operatorial properties that we discuss include nonexpansivity, firm non-expansivity, relaxed firm nonexpansivity, averagedness, being a cutter, quasi-nonexpansivity, strong quasi-nonexpansivity, strict quasi-nonexpansivity and contraction. Some relationships between operators whose componental operators have such properties and operators that have these properties on the product space are studied. This enables also to define componental fixed point sets and to study their properties. For componental contractions we offer a variant of the Banach fixed point theorem. Our motivation comes from the desire to extend a fully-simultaneous method that takes into account sparsity of the linear system in order to accelerate convergence [Censor et al., On diagonally relaxed orthogonal projection methods, SIAM J. Sci. Comput. 30 (2008), 473-504]. This was originally applicable to the linear case only and gives rise to an iterative process that uses dfferent componental operators during iterations.