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Proper splittings of Hilbert space operators

Guillermina Fongi, María Celeste Gonzalez

Abstract

Proper splittings of operators are commonly used to study the convergence of iterative processes. In order to approximate solutions of operator equations, in this article we deal with proper splittings of closed range bounded linear operators defined on Hilbert spaces. We study the convergence of general proper splittings of operators in the infinite dimensional context. We also propose some particular splittings for special classes of operators and we study different criteria of convergence and comparison for them. In some cases, these criteria are given under hypothesis of operator order relations. In addition, we relate these results with the concept of the symmetric approximation of a frame in a Hilbert space.

Proper splittings of Hilbert space operators

Abstract

Proper splittings of operators are commonly used to study the convergence of iterative processes. In order to approximate solutions of operator equations, in this article we deal with proper splittings of closed range bounded linear operators defined on Hilbert spaces. We study the convergence of general proper splittings of operators in the infinite dimensional context. We also propose some particular splittings for special classes of operators and we study different criteria of convergence and comparison for them. In some cases, these criteria are given under hypothesis of operator order relations. In addition, we relate these results with the concept of the symmetric approximation of a frame in a Hilbert space.
Paper Structure (8 sections, 37 theorems, 9 equations)

This paper contains 8 sections, 37 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proper splittings of operators and an iterative process
  4. The polar proper splitting
  5. Splittings for split operators
  6. Splittings for Hermitian operators
  7. An application to symmetric approximations of frames
  8. A few more on induced splittings

Key Result

Proposition 2.1

Consider $S,T\in\mathcal{L}(\mathcal{H})$ with closed ranges such that $ST$ has closed range. Then $(ST)^\dagger=T^\dagger S^\dagger$ if and only if $\mathcal{R}(S^*ST)\subseteq \mathcal{R}(T)$ and $\mathcal{R}(TT^*S^*)\subseteq \mathcal{R}(S^*)$.

Theorems & Definitions (82)

  • Proposition 2.1
  • Corollary 2.2
  • proof
  • Theorem 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • Definition 2.6
  • Proposition 2.7
  • Definition 3.1
  • ...and 72 more