## Abstract

In this paper we present equivalence results for several types of unbounded operator functions. A generalization of the concept equivalence after extension is introduced and used to prove equivalence and linearization for classes of unbounded operator functions. Further, we deduce methods of finding equivalences to operator matrix functions that utilizes equivalences of the entries. Finally, a method of finding equivalences and linearizations to a general case of operator matrix polynomials is presented.

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## References

Adamjan, V.M., Langer, H.: Spectral properties of a class of rational operator valued functions. J. Oper. Theory

**33**(2), 259–277 (1995)Atkinson, F.V., Langer, H., Mennicken, R., Shkalikov, A.A.: The essential spectrum of some matrix operators. Math. Nachr.

**167**, 5–20 (1994)Adamjan, V., Pivovarchik, V., Tretter, C.: On a class of non-self-adjoint quadratic matrix operator pencils arising in elasticity theory. J. Oper. Theory

**47**(2), 325–341 (2002)Bart, H., Gohberg, I., Kaashoek, M.A.: Minimal Factorization of Matrix and Operator Functions, vol. 1 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel-Boston, MA (1979)

Bart, H., Gohberg, I., Kaashoek, M.A., Ran, A.C.M.: Schur complements and state space realizations. Linear Algebra Appl.

**399**, 203–224 (2005)Bart, H., Gohberg, I., Kaashoek, M.A., Ran, A.C.M.: Factorization of Matrix and Operator Functions: The State Space Method, vol. 178 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 2008. Linear Operators and Linear Systems

den Boer, B.: Linearization of operator functions on arbitrary open sets. Integral Equ. Oper. Theory

**1**(1), 19–27 (1978)Edmunds, D.E., Evans, W.D.: Spectral theory and differential operators. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1987. Oxford Science Publications

Engström, C., Langer, H., Tretter, C.: Rational eigenvalue problems and applications to photonic crystals. J. Math. Anal. Appl.

**445**(1), 240–279 (2017)Engström, C., Torshage, A.: Enclosure of the numerical range of a class of non-selfadjoint rational operator. Integr. Equ. Oper. Theory

**88**(2), 151–184 (2017)Gohberg, I.C., Kaashoek, M.A., Lay, D.C.: Equivalence, linearization, and decomposition of holomorphic operator functions. J. Funct. Anal.

**28**(1), 102–144 (1978)Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin, 1995. Reprint of the 1980 edition

Kreĭn, M.G., Langer, H.: On some mathematical principles in the linear theory of damped oscillations of continua. I. Integral Equ. Oper. Theory, 1(3):364–399 (1978). Translated from the Russian by R. Troelstra

Kaashoek, M.A., Lunel, S.M.V.: Characteristic matrices and spectral properties of evolutionary systems. Trans. Am. Math. Soc.

**334**(2), 479–517 (1992)Kaashoek, M.A., van der Mee, C.V.M., Rodman, L.: Analytic operator functions with compact spectrum. I. Spectral nodes, linearization and equivalence. Integral Equ. Oper. Theory

**4**(4), 504–547 (1981)Markus, A.S.: Introduction to the spectral theory of polynomial operator pencils, vol. 71 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI (1988)

Mortensen, N.A., Raza, S., Wubs, M., Sondergaard, T., Bozhevolnyi, S.I.: A generalized non-local optical response theory for plasmonic nanostructures. Nat. Commun.

**5**, 3809 (2014)Nagel, R.: Towards a “matrix theory” for unbounded operator matrices. Math. Z.

**201**(1), 57–68 (1989)Nagel, R.: The spectrum of unbounded operator matrices with nondiagonal domain. J. Funct. Anal.

**89**(2), 291–302 (1990)Neven, W.H.L., Praagman, C.: Column reduction of polynomial matrices. Linear Algebra Appl.

**188**(189), 569–589 (1993)Shkalikov, A.A.: On the essential spectrum of matrix operators. Mat. Zametki

**58**(6), 945–949 (1995)Tretter, C.: A linearization for a class of \(\lambda \)-nonlinear boundary eigenvalue problems. J. Math. Anal. Appl.

**247**(2), 331–355 (2000)Tretter, C.: Boundary eigenvalue problems for differential equations \(N\eta =\lambda P\eta \) and \(\lambda \)-polynomial boundary conditions. J. Differ. Equ.

**170**(2), 408–471 (2001)Tretter, C.: Spectral Theory of Block Operator Matrices and Applications. Imperial College Press, London (2008)

## Acknowledgements

The authors gratefully acknowledge the support of the Swedish Research Council under Grant No. 621-2012-3863. We sincerely thank the reviewer for the insightful comments, which were invaluable when revising the manuscript.

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Engström, C., Torshage, A. On Equivalence and Linearization of Operator Matrix Functions with Unbounded Entries.
*Integr. Equ. Oper. Theory* **89**, 465–492 (2017). https://doi.org/10.1007/s00020-017-2415-5

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DOI: https://doi.org/10.1007/s00020-017-2415-5