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

M. Gil

Studia Mathematica (1992)

  • Volume: 103, Issue: 1, page 17-24
  • ISSN: 0039-3223

Abstract

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

How to cite

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Gil, M.. "On an estimate for the norm of a function of a quasihermitian operator." Studia Mathematica 103.1 (1992): 17-24. <http://eudml.org/doc/215931>.

@article{Gil1992,
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 $A^p - (A*)^p$ is Hilbert-Schmidt. The estimate improves one by I. Gelfand and G. Shilov.},
author = {Gil, M.},
journal = {Studia Mathematica},
keywords = {functions of linear operators; estimation of norms; quasihermitian operator; closed linear operator; convex hull of the spectrum; holomorphic function; Hilbert-Schmidt},
language = {eng},
number = {1},
pages = {17-24},
title = {On an estimate for the norm of a function of a quasihermitian operator},
url = {http://eudml.org/doc/215931},
volume = {103},
year = {1992},
}

TY - JOUR
AU - Gil, M.
TI - On an estimate for the norm of a function of a quasihermitian operator
JO - Studia Mathematica
PY - 1992
VL - 103
IS - 1
SP - 17
EP - 24
AB - 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 $A^p - (A*)^p$ is Hilbert-Schmidt. The estimate improves one by I. Gelfand and G. Shilov.
LA - eng
KW - functions of linear operators; estimation of norms; quasihermitian operator; closed linear operator; convex hull of the spectrum; holomorphic function; Hilbert-Schmidt
UR - http://eudml.org/doc/215931
ER -

References

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  1. [1] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Nauka, Moscow 1966 (in Russian). Zbl0098.30702
  2. [2] L. de Branges, Some Hilbert spaces of analytic functions, J. Math. Anal. Appl. 12 (1965), 149-186. 
  3. [3] 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. 
  4. [4] N. Dunford and J. T. Schwartz, Linear Operators, II. Spectral Theory, Selfadjoint Operators in Hilbert Space, Interscience, New York 1963. Zbl0128.34803
  5. [5] I. M. Gelfand and G. E. Shilov, Some Questions of the Theory of Differential Equations, Fiz.-Mat. Liter., Moscow 1958 (in Russian). 
  6. [6] M. I. Gil', On an estimate for the stability domain of differential systems, Differentsial'nye Uravneniya 19 (8) (1983), 1452-1454 (in Russian). Zbl0526.34039
  7. [7] 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). 
  8. [8] 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). 
  9. [9] 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. 
  10. [10] 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. 
  11. [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin 1981. Zbl0456.35001
  12. [12] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin 1966. Zbl0148.12601

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