On an estimate for the norm of a function of a quasihermitian operator

M. I. Gil

ArticleOriginal scientific text

Title

On an estimate for the norm of a function of a quasihermitian operator

Authors ¹

Affiliations

Department of Mathematics, Ben Gurion University, P.O. Box 653, Beer Sheva 84105, Israel

Abstract

Let A be a closed linear operator acting in a separable Hilbert space. Denote by co(A) the closed convex hull of the spectrum of A. An estimate for the norm of f(A) is obtained under the following conditions: f is a holomorphic function in a neighbourhood of co(A), and for some integer p the operator

A^{p} - {(A \cdot)}^{p}

is Hilbert-Schmidt. The estimate improves one by I. Gelfand and G. Shilov.

Keywords

ENG

functions of linear operators, estimation of norms

N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Nauka, Moscow 1966 (in Russian).
L. de Branges, Some Hilbert spaces of analytic functions, J. Math. Anal. Appl. 12 (1965), 149-186.
M. S. Brodskiǐ, Triangular and Jordan Representations of Linear Operators, Nauka, Moscow 1969 (in Russian); English transl.: Transl. Math. Monographs 32, Amer. Math. Soc., Providence, R.I., 1971.
N. Dunford and J. T. Schwartz, Linear Operators, II. Spectral Theory, Selfadjoint Operators in Hilbert Space, Interscience, New York 1963.
I. M. Gelfand and G. E. Shilov, Some Questions of the Theory of Differential Equations, Fiz.-Mat. Liter., Moscow 1958 (in Russian).
M. I. Gil', On an estimate for the stability domain of differential systems, Differentsial'nye Uravneniya 19 (8) (1983), 1452-1454 (in Russian).
M. I. Gil', On an estimate for the norm of a function of a Hilbert-Schmidt operator, Izv. Vyssh. Uchebn. Zaved. Mat. 1979 (8) (207), 14-19 (in Russian).
M. I. Gil', On an estimate for the resolvents of nonselfadjoint operators "close" to selfadjoint and to unitary ones, Mat. Zametki 33 (1980), 161-167 (in Russian).
I. Ts. Gokhberg and M. G. Kreǐn, Introduction to the Theory of Linear Nonselfadjoint Operators, Nauka, Moscow 1965 (in Russian); English transl.: Transl. Math. Monographs 18, Amer. Math. Soc., Providence, R.I., 1969.
I. Ts. Gokhberg and M. G. Kreǐn, Theory and Applications of Volterra Operators in Hilbert Space, Nauka, Moscow 1967 (in Russian); English transl.: Transl. Math. Monographs 24, Amer. Math. Soc., Providence, R.I., 1970.
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin 1981.
T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin 1966.

Title

On an estimate for the norm of a function of a quasihermitian operator

Affiliations

Abstract

Keywords

Bibliography