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

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

Published electronically:
February 20, 2003

MathSciNet review:
1998181

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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]**Brodskii, M.S.*Triangular and Jordan Representations of Linear Operators*, Transl. Math. Monogr., v. 32, Amer. Math. Soc. Providence, R. I. 1971. MR**48:904****[Gi1]**Gil', M.I.*Norm Estimations for Operator-valued Functions and Applications*. Marcel Dekker, Inc. New York, 1995. MR**97h:47002****[Gi2]**Gil', M.I. A nonsingularity criterion for matrices,*Linear Algebra Appl.***253**, (1997) 79-87. MR**97m:15003****[Gi3]**Gil', M.I.*Stability of Finite and Infinite Dimensional Systems*, Kluwer Ac. Publishers, Boston-Dordrecht-London, 1998. MR**99m:34110****[Gi4]**Gil', M.I. Invertibility conditions and bounds for spectra of matrix integral operators,*Monatshefte für Mathematik***129**, (2000) 15-24. MR**2000m:47062****[Gi5]**Gil', M.I. Invertibility and positive invertibility conditions of integral operators in ,*J. of Integral Equations and Appl.***13**(2001), 1-14. MR**2002c:47109****[Gi6]**Gil', M.I. On invertibility and positive invertibility of matrices,*Linear Algebra Appl.***327**, (2001), 95-104. MR**2002b:15006****[GK]**Gohberg, I. and Krein, M.G.*Theory and Applications of Volterra Operators in Hilbert Space*, Trans. Mathem. Monographs, vol. 24, Amer. Math. Soc., R.I., 1970. MR**41:9041****[Ko]**König, H.*Eigenvalue Distribution of Compact Operators*, Birkhäuser Verlag, Basel- Boston-Stuttgart, 1986. MR**88j:47021****[Kr]**Krasnosel'skii, M.A., Lifshits, J. and Sobolev A.*Positive Linear Systems. The Method of Positive Operators*, Heldermann Verlag, Berlin, 1989. MR**91f:47051****[MM]**Marcus, M. and Minc, H.*A Survey of Matrix Theory and Matrix Inequalities*, Allyn and Bacon, Boston, 1964. MR**29:112****[Pi]**Pietsch, A.*Eigenvalues and**-Numbers*, Cambridge University Press, Cambridge, 1987. MR**88j:47022b**

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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:
https://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