Singular Sylvester equation in Banach spaces and its applications: Fredholm theory approach

Djordjević, Bogdan D.

doi:10.1016/j.laa.2021.03.035

“…It was pointed out by the reviewer that our proof of (the classical, known) Proposition 2.1 contains the same expressions as those derived in [4]. Papers [2] and [3] study the case when the Sylvester operator S : X Þ Ñ AX ´XB is not invertible, but the initial equation is still solvable (with infinitely many solutions). In particular, [3] covers the case when A, B and C are scalar matrices while [2] covers the case when A, B and C are bounded linear operators in Banach spaces.…”

Section: Introductionmentioning

confidence: 83%

“…In particular, [3] covers the case when A, B and C are scalar matrices while [2] covers the case when A, B and C are bounded linear operators in Banach spaces. The results in [2] are also obtained via the Gelfand transform and spectral theory for commutative unital Banach algebras. The outline of the article is as follows.…”

Section: Introductionmentioning

confidence: 99%

The Sylvester equation in Banach algebras

Sasane¹

2021

Preprint

0

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Let A be a unital complex semisimple Banach algebra, and M A denote its maximal ideal space. For a matrix M P A nˆn , x M denotes the matrix obtained by taking entry-wise Gelfand transforms. For a matrix M P C nˆn , σpM q Ă C denotes the set of eigenvalues of M . It is shown that if A P A nˆn and B P A mˆm are such that for all ϕ P M A , σp p Apϕqq X σp p Bpϕqq " H, then for all C P A nˆm , the Sylvester equation AX ´XB " C has a unique solution X P A nˆm . As an application, Roth's removal rule is proved in the context of matrices over a Banach algebra.

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“…It was pointed out by the reviewer that our proof of (the classical, known) Proposition 2.1 contains the same expressions as those derived in [4]. Papers [2] and [3] study the case when the Sylvester operator S : X Þ Ñ AX ´XB is not invertible, but the initial equation is still solvable (with infinitely many solutions). In particular, [3] covers the case when A, B and C are scalar matrices while [2] covers the case when A, B and C are bounded linear operators in Banach spaces.…”

Section: Introductionmentioning

confidence: 86%

“…Papers [2] and [3] study the case when the Sylvester operator S : X Þ Ñ AX ´XB is not invertible, but the initial equation is still solvable (with infinitely many solutions). In particular, [3] covers the case when A, B and C are scalar matrices while [2] covers the case when A, B and C are bounded linear operators in Banach spaces. The results in [2] are also obtained via the Gelfand transform and spectral theory for commutative unital Banach algebras.…”

Section: Introductionmentioning

confidence: 99%