## Kyoto Journal of Mathematics

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

M. I. Gil’

#### Abstract

Let $H=X\otimes Y$ 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 $A=\sum_{k=0}^{m}B_{k}\otimes S^{k}$, where $B_{k}$ $(k=0,\ldots,m)$ 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

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

Dates
First available in Project Euclid: 1 August 2011

https://projecteuclid.org/euclid.kjm/1312205243

Digital Object Identifier
doi:10.1215/21562261-1299927

Mathematical Reviews number (MathSciNet)
MR2824004

Zentralblatt MATH identifier
1227.47008

#### Citation

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. https://projecteuclid.org/euclid.kjm/1312205243

#### References

• [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.