Kyoto Journal of Mathematics

Estimates for resolvents and functions of operator pencils on tensor products of Hilbert spaces

M. I. Gil’

Full-text: Open access


Let H=XY be a tensor product of separable Hilbert spaces X and Y. We establish norm estimates for the resolvent and operator-valued functions of the operator A=k=0mBkSk, where Bk (k=0,,m) are bounded operators acting in Y, and S is a self-adjoint operator acting in X. By these estimates we investigate spectrum perturbations of A. The abstract results are applied to the nonself-adjoint differential operators in Hilbert and Euclidean spaces. Our main tool is a combined use of some properties of operators on tensor products of Hilbert spaces and the recent estimates for the norm of the resolvent of a nonself-adjoint operator.

Article information

Kyoto J. Math., Volume 51, Number 3 (2011), 673-686.

First available in Project Euclid: 1 August 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A80: Tensor products of operators [See also 46M05] 47E05: Ordinary differential operators [See also 34Bxx, 34Lxx] (should also be assigned at least one other classification number in section 47) 34L15: Eigenvalues, estimation of eigenvalues, upper and lower bounds


Gil’, M. I. Estimates for resolvents and functions of operator pencils on tensor products of Hilbert spaces. Kyoto J. Math. 51 (2011), no. 3, 673--686. doi:10.1215/21562261-1299927.

Export citation


  • [1] A. Brown and C. Pearcy, Spectra of tensor products of operators, Proc. Amer. Math. Soc. 17 (1966), 162–166.
  • [2] N. Dunford and J. T. Schwartz, Linear Operators, I: General Theory. Wiley, New York, 1966.
  • [3] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1990.
  • [4] M. I. Gil’, Operator Functions and Localization of Spectra, Lecture Notes in Math. 1830, Springer, Berlin, 2003.
  • [5] M. I. Gil’, Spectrum perturbations of operators on tensor products of Hilbert spaces, J. Math. Kyoto Univ. 43 (2003), 719–735.
  • [6] M. I. Gil’, Estimates for norms of resolvents of operators on tensor products of Hilbert spaces, Period. Math. Hungar. 49 (2004), 27–41.
  • [7] M. I. Gil’, Invertibility and spectrum of operators on tensor products of Hilbert spaces, Glasg. Math. J. 46 (2004), 101–116.
  • [8] M. I. Gil’, Regular functions of operators on tensor products of Hilbert spaces, J. Integral Equations Operators 54 (2006), 317–331.
  • [9] M. I. Gil’, Norm estimates for functions of two operators on tensor products of Hilbert spaces, Math. Nachr. 281 (2008), 1–13.
  • [10] M. I. Gil’, Perturbations of operators on tensor products and spectrum localization of matrix differential operators, J. Appl. Funct. Anal. 3 (2008), 315–332.
  • [11] C. S. Kubrusly, Invariant subspaces of multiple tensor products, Acta Sci. Math. (Szeged) 75 (2009), 679–692.
  • [12] C. S. Kubrusly and B. P. Duggal, On Weyl and Browder spectra of tensor products, Glasg. Math. J. 50 (2008), 289–302.
  • [13] C. S. Kubrusly and P. C. M. Vieira, Convergence and decomposition for tensor products of Hilbert space operators, Oper. Matrices 2 (2008), 407–416.
  • [14] P. Kurasov and S. Naboko, On the essential spectrum of a class of singular matrix differential operators, I: Quasi-regularity conditions and essential self-adjointness, Math. Phys. Anal. Geom. 5 (2002), 243–286.
  • [15] J. Locker, Spectral Theory of Nonself-adjoint Two Point Differential Operators, Amer. Math. Soc. Math. Surveys Monogr. 73, Providence, 1999.
  • [16] C. M. Carracedo and M. S. Alix, The Theory of Fractional Powers of Operators, Elsevier, Amsterdam, 2001.
  • [17] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, 2nd ed., Academic Press, New York, 1980.
  • [18] R. A. Ryan, Introduction to Tensor Products of Banach Spaces, Springer, Berlin, 2002.
  • [19] L. A. Shuster, Estimates of eigenfunctions and localization of the spectrum of differential operators, J. Math. Anal. Appl. 229 (1999), 363–375.
  • [20] J. Weidmann, Spectral Theory of Differential Operators, Springer, Berlin, 1987.