Abstract
Let Z 0 be a bounded operator in a Banach space X with purely essential spectrum and K a nuclear operator in X. We construct a holomorphic function the zeros of which coincide with the discrete spectrum of Z 0+K and derive a Lieb-Thirring type inequality. We obtain estimates for the number of eigenvalues in certain regions of the complex plane and an estimate for the asymptotics of the eigenvalues approaching to the essential spectrum of Z 0.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. A. Abramov, A. Aslanyan and E. B. Davies, Bounds on complex eigenvalues and resonances, J. Phys. A, 34(1) (2001), 57–72.
A. Borichev, L. Golinskii and S. Kupin, A Blaschke-Type condition and its application to complex Jacobi matrices, Bull. London Math. Soc., 41 (2009), 117–123.
V. Bruneau and E. M. Ouhabaz, Lieb-Thirring estimates for non-self-adjoint Schrödinger operators, J.Math. Phys., 49(9) 093504 (2008).
E. B. Davies, Linear operators and their spectra, Cambridge University Press, Cambridge (2008).
E. B. Davies and J. Nath, Schrödinger operators with slowly decaying potentials, On the occasion of the 65th birthday of Professor Michael Eastham, J. Comput. Appl. Math., 148(1) (2002), 1–28.
M. Demuth, M. Hansmann and G. Katriel, Eigenvalues of non-selfadjoint operators: a comparison of two approaches, Operator Theory: Advances and Applications, 232 (2013), 107–163.
P. L. Duren, On the spectrum of a Toeplitz operator, Pacific J. Math., 14 (1964), 21–29.
S. Favorov and L. Golinskii, Blaschke-type conditions in unbounded domains, generalized convexity and applications in perturbation theory, Rev. Mat. Iberoam, 31(1) (2015), 1–32.
R. L. Frank, Eigenvalue bounds for Schrödinger operators with complex potentials, Bull. Lond. Math. Soc., 43(4) (2011), 745–750.
R. L. Frank, A. Laptev, E. H. Lieb and R. Seiringer, Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials, Lett. Math. Phys., 77(3) (2006), 309–316.
I. C. Gohberg, S. Goldberg and M. A. Kaashoek, Classes of linear operators Vol. 1., Birkhäuser Verlag, Basel (1990).
I. C. Gohberg, S. Goldberg and N. Krupnik, Traces and determinants of linear operators, Birkhäuser Verlag, Basel (2000).
L. Golinskii and S. Kupin, Lieb-Thirring bounds for complex Jacobi matrices, Lett. Math. Phys., 82(1) (2007), 79–90.
M. Hansmann, On the discrete spectrum of linear operators in Hilbert spaces, Dissertation, TU Clausthal, 2010. See http://nbn-resolvin.de/urn:nbn:de:gbv:1041097281 for an electronic version.
M. Hansmann, An eigenvalue estimate and its application to non-selfadjoint Jacobi and Schrödinger operators, Lett. Math. Phys., 98(1) (2011), 79–95.
M. Hansmann, Variation of discrete spectra for non-selfadjoint perturbations of selfadjoint operators, Integral Equations Operator Theory, 76 (2013), 163–178.
M. Hansmann and G. Katriel, Inequalities for the eigenvalues of non-selfadjoint Jacobi operators, Complex Anal. Oper. Theory, 5(1) (2011), 197–218.
J. S. Howland, Analyticity of determinants of operators on a banach space, Proc. Amer. Math. Soc., 1(28) (1971), 177–180.
B. Jacob, Zeros of Fredholm operator valued Hp-functions, Math. Nachr, 227 (2001), 81–97.
T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin (1995).
A. Laptev and O. Safronov, Eigenvalue estimates for Schrödinger operators with complex valued potentials, Comm. Math. Phys., 292(1) (2009), 29–54.
R. Nevanlinna, Eindeutige analytische Funktionen, Springer-Verlag, Berlin (1953).
A. Pietsch, Eigenvalues and s-numbers, Cambrdige University Press, Cambridge (1987).
R. Remmert and G. Schumacher, Funktionentheorie I. Springer, Berlin - Heidelberg, 6. edition, (2002).
W. Rudin, Real and complex analysis, McGraw-Hill book Co., New York, third edition, (1987).
O. Safronov, Estimates for eigenvalues of the Schrödinger operator with a complex potential, Bull. Lond. Math. Soc., 42(3) (2010), 452–456.
B. Simon, Notes on infinite determinants of Hilbert space operators, Advances in Math., 24(3) (1977), 244–273.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Kalyan B. Sinha on the occasion of his 70th birthday.
Rights and permissions
About this article
Cite this article
Demuth, M., Hanauska, F. On the distribution of the discrete spectrum of nuclearly perturbed operators in Banach spaces. Indian J Pure Appl Math 46, 441–462 (2015). https://doi.org/10.1007/s13226-015-0145-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13226-015-0145-4