Advances in Operator Theory

Norm estimates for resolvents of linear operators in a Banach space and spectral variations

Michael Gil'

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $P_t$ $(a\le t\le b)$ be a function whose values are projections in a Banach space. The paper is devoted to bounded linear operators $A$ admitting the representation $$A=\int_a^b \phi(t)dP_{t}+V,$$ where $\phi(t)$ is a scalar function and $V$ is a compact quasi-nilpotent operator such that $P_tVP_t=VP_t$ $(a\le t\le b)$. We obtain norm estimates for the resolvent of $A$ and a bound for the spectral variation of $A$. In addition, the representation for the resolvents of the considered operators is established via multiplicative operator integrals. That representation can be considered as a generalization of the representation for the resolvent of a normal operator in a Hilbert space. It is also shown that the considered operators are Kreiss-bounded. Applications to integral operators in $L^p$ are also discussed. In particular, bounds for the upper and lower spectral radius of integral operators are derived.

Article information

Source
Adv. Oper. Theory, Volume 4, Number 1 (2019), 113-139.

Dates
Received: 11 January 2018
Accepted: 13 April 2018
First available in Project Euclid: 27 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.aot/1524816021

Digital Object Identifier
doi:10.15352/aot.1801-1293

Mathematical Reviews number (MathSciNet)
MR3867337

Zentralblatt MATH identifier
06946446

Subjects
Primary: 47A10: Spectrum, resolvent
Secondary: 47A55: Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15] 47G10: Integral operators [See also 45P05] 47A11: Local spectral properties 47A30: Norms (inequalities, more than one norm, etc.)

Keywords
Banach space resolvent spectral variation integral operator invariant chain of projections

Citation

Gil', Michael. Norm estimates for resolvents of linear operators in a Banach space and spectral variations. Adv. Oper. Theory 4 (2019), no. 1, 113--139. doi:10.15352/aot.1801-1293. https://projecteuclid.org/euclid.aot/1524816021


Export citation

References

  • R. Bhatia, Perturbation bounds for matrix eigenvalues, Reprint of the 1987 original. Classics in Applied Mathematics, 53. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007.
  • L. de Branges, Some Hilbert spaces of analytic functions II, J. Math. Anal. Appl. 11 (1965), 44–72.
  • M. S. Brodskii, Triangular and Jordan representations of linear operators, Translated from the Russian by J. M. Danskin. Translations of Mathematical Monographs, Vol. 32. American Mathematical Society, Providence, R.I., 1971.
  • V. M. Brodskii, I. C. Gohberg, and M. G. Krein, General theorems on triangular representations of linear operators and multiplicative representations of their characteristic functions, (Russian) Funkcional. Anal. i Priložen. 3 (1969), no. 4, 1–27.
  • J. Diestel, H. Jarchow, and A. Tonge, Absolutely summing operators, Cambridge Studies in Advanced Mathematics, 43. Cambridge University Press, Cambridge, 1995.
  • N. Dunford and J. T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle Interscience Publishers John Wiley & Sons New York-London 1963
  • N. Dunford and J. T. Schwartz, Linear operators. Part III: Spectral operators, With the assistance of William G. Bade and Robert G. Bartle. Pure and Applied Mathematics, Vol. VII. Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1971.
  • S. P. Eveson, Norms of iterates of Volterra operators on $L^2$. J. Operator Theory 50 (2003), no. 2, 369–386.
  • S. P. Eveson, Asymptotic behaviour of iterates of Volterra operators on $L^p(0, 1)$. Integral Equations Operator Theory 53 (2005), 331–-341.
  • M. I. Gil', On the representation of the resolvent of a nonselfadjoint operator by the integral with respect to a spectral function, Soviet Math. Dokl. 14 (1973), 1214–1217.
  • M. I. Gil', One estimate for resolvents of nonselfadjoint operators which are "near" to selfadjoint and to unitary ones, Math. Notes 33 (1983), 81–84.
  • M. I. Gil', Operator functions and localization of spectra, Lecture Notes in Mathematics, 1830. Springer-Verlag, Berlin, 2003.
  • M. I. Gil', Kronecker's products and Kronecker's sums of operators, Contributions in mathematics and engineering, 205–253, Springer, [Cham], 2016.
  • M. I. Gil', Operator functions and operator equations, World Scientific, New Jersey, 2017.
  • I. C. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of linear operators. Vol. II, Operator Theory: Advances and Applications, 63. Birkhäuser Verlag, Basel, 1993.
  • I. C. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18 American Mathematical Society, Providence, R.I. 1969
  • I. C. Gohberg and M. G. Krein, 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.
  • D. Kershaw, Operator norms of powers of the Volterra operator, J. Integral Equations Appl. 11 (1999), no. 3, 351–362.
  • N. Lao and R. Whitley, Norms of powers of the Volterra operator, Integral Equations Operator Theory 27 (1997), no. 4, 419–425
  • G. Little and J. B. Reade, Estimates for the norm of the $n$th indefinite integral, Bull. London Math. Soc. 30 (1998), no. 5, 539–542.
  • A. Montes-Rodriguez, J. Sanchez-Alvarez, and J. Zemanek, Uniform Abel–Kreiss boundedness and the extremal behaviour of the Volterra operator, Proc. London Math. Soc. (3)91 (2005), no. 3, 761–788.
  • A. Pietsch, Eigenvalues and $s$-numbers, Cambridge Studies in Advanced Mathematics, 13. Cambridge University Press, Cambridge, 1987.
  • H. Radjavi and P. Rosenthal, Invariant subspaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77. Springer-Verlag, New York-Heidelberg, 1973.
  • L. Sakhnovich, $(S + N)$-triangular operators: spectral properties and important examples, Math. Nachr. 289 (2016), no. 13, 1680–-1691.
  • G. W. Stewart and J. G. Sun, Matrix perturbation theory, Computer Science and Scientific Computing. Academic Press, Inc., Boston, MA, 1990.
  • J.C. Strikwerda and B. A. Wade, A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions, Linear operators (Warsaw, 1994), 339–360, Banach Center Publ., 38, Polish Acad. Sci. Inst. Math., Warsaw, 1997.
  • B. Thorpe, The norm of powers of the indefinite integral operator on $(0, 1)$, Bull. London Math. Soc. 30 (1998), no. 5, 543–548.