On Componental Operators in Hilbert Space. (3rd October 2021)

• Record Type: Journal Article
• Authors: Cegielski, Andrzej; Censor, Yair
• Is Part Of: Numerical functional analysis and optimization. Volume 42:Number 13(2021)
• Page Start: 1555
• Page End: 1571
• Publication Date: 2021-10-03
• Subjects: Componental operators -- fixed point -- Hilbert spaces -- nonexpansive operator -- quasi-nonexpansive operator; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2021.2006695
• Languages: English
• ISSNs: 0163-0563

Abstract

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 nonexpansivity, 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. This was originally applicable to the linear case only and gives rise to an iterative process that uses different componental operators during iterations.