Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions

Keywords: Generalized Nevanlinna function; Linear relation; Operator representation; Jordan chain

Abstract

UDC 517.9
We present necessary and sufficient conditions for the reducibility of a self-adjoint linear
relation in a Krein space.
Then a generalized Nevanlinna function $Q$ represented by a self-adjoint linear
relation $A$ in a Pontryagin space is decomposed by means of the reducing subspaces of $A.$
The sum of two functions $Q_{i}{\in N}_{\kappa_{i}}(\mathcal{H}),$ $i=1, 2,$ minimally
represented by the triplets $(\mathcal{K}_{i},A_{i},\Gamma_{i})$ is also studied.
For this purpose, we create a model $(\tilde{\mathcal{K}},\tilde{A},\tilde{\Gamma })$ to
represent $Q:=Q_{1}+Q_{2}$ in terms of $(\mathcal{K}_{i},A_{i},\Gamma_{i})$.
By using this model, necessary and sufficient conditions for $\kappa =\kappa_{1}+\kappa_{2}$ are proved in the analytic form.
Finally, we explain how degenerate Jordan chains of the representing relation $A$ affect the reducing subspaces of $A$ and the
decomposition of the corresponding function $Q.$

References

R. Arens, Operational calculus of linear relations, Pacif. J. Math., 11, 9 – 23 (1961). DOI: https://doi.org/10.2140/pjm.1961.11.9

N. I. Akhiezer, I. M. Glazman, Theory of linear operators in Hilbert space, Dover Publ., Inc. (1993).

D. Alpay, A. Dijksma, J. Rovnyak, H. de Snoo, Schur functions, operator colligations, and reproducing kernel pontryagin spaces, Oper. Theory Adv. and Appl., 96 (1997), https://doi.org/10.1007/978-3-0348-8908-7 DOI: https://doi.org/10.1007/978-3-0348-8908-7

J. Behrndt, H. de Snoo, S. Hassi, Boundary value problems, Weyl functions, and differential operators, Monographs Math., 108 (2020); https://doi.org/10.1007/978-3-030-36714-5. DOI: https://doi.org/10.1007/978-3-030-36714-5

J. Bognar, Indefinite inner product spaces, Springer-Verlag, Berlin etc. (1974). DOI: https://doi.org/10.1007/978-3-642-65567-8

M. Borogovac, Inverse of generalized Nevanlinna function that is holomorphic at infinity, North-West Eur. J. Math., 6, 19 – 43 (2020), https://doi.org/10.32014/2019.2518-1726.72 DOI: https://doi.org/10.32014/2019.2518-1726.72

M. Borogovac, H. Langer, A characterization of generalized zeros of negative type of matrix functions of the class $N^{ntimes n}_kappa$, Oper. Theory Adv. and Appl., 28, 17 – 26 (1988). DOI: https://doi.org/10.1007/978-3-0348-9164-6_1

K. Daho, H. Langer, Matrix functions of the class $N_kappa$ Math. Nachr., 120, 275 – 294 (1985), https://doi.org/10.1002/mana.19851200123 DOI: https://doi.org/10.1002/mana.19851200123

A. Dijksma, H. Langer, H. S. V. de Snoo, Eigenvalues and pole functions of Hamiltonian systems with eigenvalue depending boundary conditions, Math. Nachr., 161, 107 – 154 (1993), https://doi.org/10.1002/mana.19931610110 DOI: https://doi.org/10.1002/mana.19931610110

S. Hassi, H. S. V. de Snoo, H. Woracek, Some interpolation problems of Nevanlinna – Pick type, Oper. Theory Adv. and Appl., 106, 201 – 216 (1998). DOI: https://doi.org/10.1007/978-3-0348-8812-7_10

I. S. Iohvidov, M. G. Kreĭn, H. Langer, Introduction to the spectral theory of operators in spaces with an indefinite metric, Akad.-Verlag, Berlin (1982).

M. G. Kreĭn, H. Langer, Über die $Q$-Funktion eines $pi $-hermiteschen Operators im Raume $Pi sb{kappa }$. (German), Acta Sci. Math., 34, 190 – 230 (1973).

M. G. Kreĭn, H. Langer, Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume $Pi sb{kappa }$ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr., 77, 187 – 236 (1977), https://doi.org/10.1002/mana.19770770116 DOI: https://doi.org/10.1002/mana.19770770116

H. Langer, B. Textorius, On generalized resolvents and $Q$-functions of symmetric linear relations (subspaces) in Hilbert space, Pacif. J. Math., 72, № 1, 135 – 165 (1977). DOI: https://doi.org/10.2140/pjm.1977.72.135

A. Luger, Generalized Nevanlinna functions: operator representations, asymptotic behavior, Operator Theory (2014), p. 345 – 371; https://doi.org/10.1007/978-3-0348-0667-1-35. DOI: https://doi.org/10.1007/978-3-0348-0667-1_35

A. Luger, A characterization of generalized poles of generalized Nevanlinna functions, Math. Nachr., 279, 891 – 910 (2006), https://doi.org/10.1002/mana.200310401 DOI: https://doi.org/10.1002/mana.200310401

H. de Snoo, H. Woracek, The Krein formula in almost Pontryagin spaces. A proof via orthogonal coupling, Indag. Math. (N. S.), 29, № 2, 714 – 729 (2018), https://doi.org/10.1016/j.indag.2017.12.001 DOI: https://doi.org/10.1016/j.indag.2017.12.001

P. Sorjonen, On linear relations in an indefinite inner product space, Ann. Acad. Sci. Fenn. Math., 4, 169 – 192 (1978/1979), https://doi.org/10.5186/aasfm.1978-79.0424 DOI: https://doi.org/10.5186/aasfm.1978-79.0424

Published
09.08.2022
How to Cite
BorogovacM. “Reducibility of Self-Adjoint Linear Relations and Application to Generalized Nevanlinna Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 7, Aug. 2022, pp. 893 -20, doi:10.37863/umzh.v74i7.6084.
Section
Research articles