On the operator equation A X − X B = Cwith unbounded operators A, B, and C

In this paper we solve Sylvester matrix equation with infinitely-many solutions and conduct their classification. If the conditions for their existence are not met, we provide a way for their approximation by least-squares minimal-norm method. with solution X ∈ L(V 1 , V 2) are called Sylvester equations, Sylvester-Rosenblum equations or algebraic Ricatti equations. Such equations have various application in vast fields of mathematics, physics, computer science and engineering (see e.g. [5], [19] and references therein). Fundamental results, established by Sylvester and Rosenblum themselves, are now-days the starting point in solving contemporary problems where these equations occur. These results are Theorem 1.1. [23] (Sylvester matrix equation) Let A, B and C be matrices. Equation AX − XB = C has unique solution X iff σ(A) ∩ σ(B) = ∅. Theorem 1.2. [22] (Rosenblum operator equation) Let A, B and C be bounded linear operators. Equation AX − XB = C has unique solution X if σ(A) ∩ σ(B) = ∅. Equations with unique solutions have been extensively studied so far. There are numerous results regarding this case, some of them theoretical (e. g. Lyapunov stability criteria and spectral operators), which can be found in [2], [5] or [9], and some of them computational (matrix sign function, factorization of matrices and operators, various iterative methods etc.). It should be mentioned that matrix eq. (1) with unique solution X has been solved numerically (among others

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“…For given vector spaces A, B and C are unbounded operators but solution X is unique and bounded has been studied in [16] and [20].…”

Section: Introductionmentioning

confidence: 99%

Classification and approximation of solutions to Sylvester matrix equation

Djordjevic

Dincic

2019

Filomat

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“…The Sylvester equation (37) with unbounded operator coefficients A and B is considered in [2,3,39,79,94,104,116].…”

Section: The Impulse Responsementioning

confidence: 99%

Analytic functional calculus for two operators

Kurbatov

Kurbatova

Oreshina

2016

Preprint

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Properties of the mappingsλ dλ are discussed; here R 1, (•) and R 2, (•) are pseudo-resolvents, i. e., resolvents of bounded, unbounded, or multivalued linear operators, and f and g are analytic functions. Several applications are considered: a representation of the impulse response of a second order linear differential equation with operator coefficients, a representation of the solution of the Sylvester equation, and an exploration of properties of the differential of the ordinary functional calculus.

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