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Triangular representations of functions of operators with Schatten–von Neumann Hermitian components
Author(s) -
Gil' Michael
Publication year - 2020
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800098
Subject(s) - mathematics , multiplication operator , quasinormal operator , von neumann's theorem , operator norm , finite rank operator , compact operator , hermitian matrix , hilbert space , bounded operator , operator (biology) , pure mathematics , norm (philosophy) , shift operator , separable space , operator space , unitary operator , resolvent , bounded function , mathematical analysis , banach space , biochemistry , chemistry , repressor , computer science , transcription factor , law , extension (predicate logic) , political science , gene , programming language
Let H be a separable Hilbert space with the unit operator I , let A be a bounded linear operator in H with a Schatten–von Neumann Hermitian component ( A −A ∗ ) / 2 i ( A ∗ means the operator adjoint to A ) and let f ( z ) be a function analytic on the spectra of A and A ∗ . For f ( A ) we obtain the representation in the form of the sum of a normal operator and a quasi‐nilpotent Schatten–von Neumann operator Vf , and estimate the norm of Vf . That estimate gives us an inequality for the norm of the resolvent( λ I − f ( A ) ) − 1of f ( A )( λ ∈ C ) . Applications of the obtained estimate for( λ I − f ( A ) ) − 1to operator equations, whose coefficients are operator functions, and to perturbations of spectra are also discussed.