Triangular representations of functions of operators with Schatten–von Neumann Hermitian components

Michael Gil'

Mathematische Nachrichten, 2020, vol. 293, issue 10, 1947-1960

Abstract: 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∗)/2i (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 (λI−f(A))−1 of f(A) (λ∈C). Applications of the obtained estimate for (λI−f(A))−1 to operator equations, whose coefficients are operator functions, and to perturbations of spectra are also discussed.

Date: 2020
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