Inner bounds for the spectrum of quasinormal operators

Author:
M. I. Gil'

Journal:
Proc. Amer. Math. Soc. **131** (2003), 3737-3746

MSC (2000):
Primary 47A55, 47A75; Secondary 47G10, 47G20

Published electronically:
February 20, 2003

MathSciNet review:
1998181

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Abstract | References | Similar Articles | Additional Information

Abstract: A linear operator in a separable Hilbert space is called a quasinormal one if it is a sum of a normal operator and a compact one. In the paper, bounds for the spectrum of quasinormal operators are established. In addition, the lower estimate for the spectral radius is derived. Under some restrictions, that estimate improves the well-known results. Applications to integral operators and matrices are discussed. Our results are new even in the finite-dimensional case.

**[Br]**M. S. Brodskiĭ,*Triangular and Jordan representations of linear operators*, American Mathematical Society, Providence, R.I., 1971. Translated from the Russian by J. M. Danskin; Translations of Mathematical Monographs, Vol. 32. MR**0322542****[Gi1]**Michael I. Gil′,*Norm estimations for operator-valued functions and applications*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 192, Marcel Dekker, Inc., New York, 1995. MR**1352684****[Gi2]**M. I. Gil,*A nonsingularity criterion for matrices*, Linear Algebra Appl.**253**(1997), 79–87. MR**1431166**, 10.1016/0024-3795(95)00704-0**[Gi3]**Michael I. Gil′,*Stability of finite- and infinite-dimensional systems*, The Kluwer International Series in Engineering and Computer Science, 455, Kluwer Academic Publishers, Boston, MA, 1998. MR**1666431****[Gi4]**M. I. Gil′,*Invertibility conditions and bounds for spectra of matrix integral operators*, Monatsh. Math.**129**(2000), no. 1, 15–24. MR**1741036**, 10.1007/s006050050002**[Gi5]**M. I. Gil’,*Invertibility and positive invertibility of integral operators in 𝐿^{∞}*, J. Integral Equations Appl.**13**(2001), no. 1, 1–14. MR**1827372**, 10.1216/jiea/996986880**[Gi6]**M. I. Gil′,*On invertibility and positive invertibility of matrices*, Linear Algebra Appl.**327**(2001), no. 1-3, 95–104. MR**1823342**, 10.1016/S0024-3795(00)00313-X**[GK]**I. C. Gohberg and M. G. Kreĭn,*Theory and applications of Volterra operators in Hilbert space*, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 24, American Mathematical Society, Providence, R.I., 1970. MR**0264447****[Ko]**Hermann König,*Eigenvalue distribution of compact operators*, Operator Theory: Advances and Applications, vol. 16, Birkhäuser Verlag, Basel, 1986. MR**889455****[Kr]**M. A. Krasnosel′skij, Je. A. Lifshits, and A. V. Sobolev,*Positive linear systems*, Sigma Series in Applied Mathematics, vol. 5, Heldermann Verlag, Berlin, 1989. The method of positive operators; Translated from the Russian by Jürgen Appell. MR**1038527****[MM]**Marvin Marcus and Henryk Minc,*A survey of matrix theory and matrix inequalities*, Allyn and Bacon, Inc., Boston, Mass., 1964. MR**0162808****[Pi]**Albrecht Pietsch,*Eigenvalues and 𝑠-numbers*, Cambridge Studies in Advanced Mathematics, vol. 13, Cambridge University Press, Cambridge, 1987. MR**890520**

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Additional Information

**M. I. Gil'**

Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel

Email:
gilmi@black.bgu.ac.il

DOI:
http://dx.doi.org/10.1090/S0002-9939-03-06950-8

Keywords:
Linear operators,
spectrum,
spectral radius,
integral operators,
finite and infinite matrices

Received by editor(s):
March 21, 2001

Received by editor(s) in revised form:
June 24, 2002

Published electronically:
February 20, 2003

Additional Notes:
This research was supported by the Israel Ministry of Science and Technology

Communicated by:
Joseph A. Ball

Article copyright:
© Copyright 2003
American Mathematical Society