Inner bounds for the spectrum of quasinormal operators
Author:
M. I. Gil'
Journal:
Proc. Amer. Math. Soc. 131 (2003), 37373746
MSC (2000):
Primary 47A55, 47A75; Secondary 47G10, 47G20
Published electronically:
February 20, 2003
MathSciNet review:
1998181
Fulltext PDF Free Access
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 wellknown results. Applications to integral operators and matrices are discussed. Our results are new even in the finitedimensional 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
(48 #904)
 [Gi1]
Michael
I. Gil′, Norm estimations for operatorvalued functions and
applications, Monographs and Textbooks in Pure and Applied
Mathematics, vol. 192, Marcel Dekker, Inc., New York, 1995. MR 1352684
(97h:47002)
 [Gi2]
M.
I. Gil, A nonsingularity criterion for matrices, Linear
Algebra Appl. 253 (1997), 79–87. MR 1431166
(97m:15003), http://dx.doi.org/10.1016/00243795(95)007040
 [Gi3]
Michael
I. Gil′, Stability of finite and infinitedimensional
systems, The Kluwer International Series in Engineering and Computer
Science, 455, Kluwer Academic Publishers, Boston, MA, 1998. MR 1666431
(99m:34110)
 [Gi4]
M.
I. Gil′, Invertibility conditions and bounds for spectra of
matrix integral operators, Monatsh. Math. 129 (2000),
no. 1, 15–24. MR 1741036
(2000m:47062), http://dx.doi.org/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
(2002c:47109), http://dx.doi.org/10.1216/jiea/996986880
 [Gi6]
M.
I. Gil′, On invertibility and positive invertibility of
matrices, Linear Algebra Appl. 327 (2001),
no. 13, 95–104. MR 1823342
(2002b:15006), http://dx.doi.org/10.1016/S00243795(00)00313X
 [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
(41 #9041)
 [Ko]
Hermann
König, Eigenvalue distribution of compact operators,
Operator Theory: Advances and Applications, vol. 16, Birkhäuser
Verlag, Basel, 1986. MR 889455
(88j:47021)
 [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
(91f:47051)
 [MM]
Marvin
Marcus and Henryk
Minc, A survey of matrix theory and matrix inequalities, Allyn
and Bacon, Inc., Boston, Mass., 1964. MR 0162808
(29 #112)
 [Pi]
Albrecht
Pietsch, Eigenvalues and 𝑠numbers, Cambridge Studies
in Advanced Mathematics, vol. 13, Cambridge University Press,
Cambridge, 1987. MR 890520
(88j:47022b)
 [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 Operatorvalued 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) 7987. MR 97m:15003
 [Gi3]
 Gil', M.I. Stability of Finite and Infinite Dimensional Systems, Kluwer Ac. Publishers, BostonDordrechtLondon, 1998. MR 99m:34110
 [Gi4]
 Gil', M.I. Invertibility conditions and bounds for spectra of matrix integral operators, Monatshefte für Mathematik 129, (2000) 1524. MR 2000m:47062
 [Gi5]
 Gil', M.I. Invertibility and positive invertibility conditions of integral operators in , J. of Integral Equations and Appl. 13 (2001), 114. MR 2002c:47109
 [Gi6]
 Gil', M.I. On invertibility and positive invertibility of matrices, Linear Algebra Appl. 327, (2001), 95104. 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 BostonStuttgart, 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
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
47A55,
47A75,
47G10,
47G20
Retrieve articles in all journals
with MSC (2000):
47A55,
47A75,
47G10,
47G20
Additional Information
M. I. Gil'
Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, BeerSheva 84105, Israel
Email:
gilmi@black.bgu.ac.il
DOI:
http://dx.doi.org/10.1090/S0002993903069508
PII:
S 00029939(03)069508
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
