Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Solvability of The Operator Equations $ AX-XB=C $ and $ AX-YB=C $

Farida Lombarkia, Assia Bezai, Néstor Thome

TL;DR

This work extends classical Roth–Rosenblum solvability results for operator equations to the setting of bounded operators on infinite-dimensional Hilbert spaces by leveraging group inverses and generalized inverses. It introduces the notions of pseudo-similarity and pseudo-equivalence to relate block operator matrices $M=AC0B$ and $D=A00B$, establishing necessary and sufficient conditions for solvability of $AX-XB=C$ and $AX-YB=C$, and providing explicit parametrized solution formulas. For the Stein-type equation $AYB-Y=C$, the authors derive equivalent pseudo-relations and invertibility criteria that yield constructive solution descriptions. Overall, the results generalize classical matrix theory to a functional-analytic context and supply practical, explicit solution characterizations in terms of $A^{\sharp}$, $B^{\sharp}$, and spectral projections $A^{\pi}$, $B^{\pi}$.

Abstract

This paper provides new necessary and sufficient conditions for the solvability to the operator equations $ AX-XB=C$ and $AX-YB=C,$ where $A $ and $B $ are group invertible operators defined on an infinite dimensional Hilbert space. In addition, the general solutions to the equation $AX-YB=C,$ are derived in terms of group inverse of $ A $ and $ B $. As a consequence, new necessary and sufficient conditions for the solvability to the operator equation $ AYB-Y=C,$ are derived.

Solvability of The Operator Equations $ AX-XB=C $ and $ AX-YB=C $

TL;DR

This work extends classical Roth–Rosenblum solvability results for operator equations to the setting of bounded operators on infinite-dimensional Hilbert spaces by leveraging group inverses and generalized inverses. It introduces the notions of pseudo-similarity and pseudo-equivalence to relate block operator matrices and , establishing necessary and sufficient conditions for solvability of and , and providing explicit parametrized solution formulas. For the Stein-type equation , the authors derive equivalent pseudo-relations and invertibility criteria that yield constructive solution descriptions. Overall, the results generalize classical matrix theory to a functional-analytic context and supply practical, explicit solution characterizations in terms of , , and spectral projections , .

Abstract

This paper provides new necessary and sufficient conditions for the solvability to the operator equations and where and are group invertible operators defined on an infinite dimensional Hilbert space. In addition, the general solutions to the equation are derived in terms of group inverse of and . As a consequence, new necessary and sufficient conditions for the solvability to the operator equation are derived.
Paper Structure (4 sections, 6 theorems, 41 equations)

This paper contains 4 sections, 6 theorems, 41 equations.

Table of Contents

  1. Introduction and basic definitions
  2. Pseudo-similarity and pseudo-equivalence of operators
  3. Solvability of the operator equation $AX-XB=C$
  4. The operator equation $AX-YB = C$

Key Result

Theorem 2.3

Let $A \in\mathcal{B}(\mathcal{H})$ and $B \in\mathcal{B}(\mathcal{K})$, suppose that $A$ is pseudo-similar to $B$ via $T \in\mathcal{B}(\mathcal{K},\mathcal{H})$ and $T^{-}$, $T^{=}$ are as in Definition def1. Then the following are valid and

Theorems & Definitions (16)

  • Definition 2.1
  • Example 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Theorem 3.1
  • proof
  • ...and 6 more