Compressed resolvents of selfadjoint contractive exit space extensions and holomorphic operator-valued functions associated with them

Abstract

Contractive selfadjoint extensions of a Hermitian contraction $B$ in a Hilbert space $\mathfrak H$ with an exit in some larger Hilbert space $\mathfrak H\oplus\mathcal H$ are investigated. This leads to a new geometric approach for characterizing analytic properties of holomorphic operator-valued functions of Krein-Ovcharenko type, a class of functions whose study has been recently initiated by the authors. Compressed resolvents of such exit space extensions are also investigated leading to some new connections to transfer functions of passive discrete-time systems and related classes of holomorphic operator-valued functions.

Key words: Selfadjoint extension, compressed resolvent, transfer function.

Full Text

Article Information

Title	Compressed resolvents of selfadjoint contractive exit space extensions and holomorphic operator-valued functions associated with them
Source	Methods Funct. Anal. Topology, Vol. 21 (2015), no. 3, 199-224
MathSciNet	MR3521692
zbMATH	06630268
Milestones	Received 01/02/2015; Revised 20/02/2015
Copyright	The Author(s) 2015 (CC BY-SA)

Authors Information

Yu. M. Arlinskiĭ
Department of Mathematical Analysis, East Ukrainian National University; Department of Mathematics, Dragomanov National Pedagogical University, 9 Pirogova Str., Kyiv, 01601, Ukraine

S. Hassi
Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, 65101 Vaasa, Finland

Citation Example

Yu. M. Arlinskiĭ and S. Hassi, Compressed resolvents of selfadjoint contractive exit space extensions and holomorphic operator-valued functions associated with them, Methods Funct. Anal. Topology 21 (2015), no. 3, 199-224.

BibTex

@article {MFAT777,
    AUTHOR = {Arlinskiĭ, Yu. M. and Hassi, S.},
     TITLE = {Compressed resolvents of selfadjoint contractive exit space extensions and holomorphic operator-valued functions associated with them},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {21},
      YEAR = {2015},
    NUMBER = {3},
     PAGES = {199-224},
      ISSN = {1029-3531},
  MRNUMBER = {MR3521692},
 ZBLNUMBER = {06630268},
       URL = {http://mfat.imath.kiev.ua/article/?id=777},
}

References

1. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Spaces, Monographs and Studies in Mathematics, Vol. 9, 10, Pitman Advanced Publishing Program, Boston-London-Melbourne, 1981.
2. William N. Anderson Jr., Shorted operators, SIAM J. Appl. Math. 20 (1971), 520-525.  MathSciNet
3. W. N. Anderson Jr., R. J. Duffin, Series and parallel addition of matrices, J. Math. Anal. Appl. 26 (1969), 576-594.  MathSciNet
4. W. N. Anderson Jr., G. E. Trapp, Shorted operators. II, SIAM J. Appl. Math. 28 (1975), 60-71.  MathSciNet
5. Yu. M. Arlinskii, A class of contractions in Hilbert space, Ukrain. Mat. Zh. 39 (1987), no. 6, 691-696, 813.  MathSciNet
6. Yu. M. Arlinskii, Characteristic functions of operators of the class $C(\alpha)$, Izv. Vyssh. Uchebn. Zaved. Mat. (1991), no. 2, 13-21.  MathSciNet
7. Yury Arlinskii, The Kalman-Yakubovich-Popov inequality for passive discrete time-invariant systems, Oper. Matrices 2 (2008), no. 1, 15-51.  MathSciNet CrossRef
8. Yuri Arlinskii, Sergey Belyi, Eduard Tsekanovskii, Conservative realizations of Herglotz-Nevanlinna functions, Birkhauser/Springer Basel AG, Basel, 2011.  MathSciNet CrossRef
9. Yuri M. Arlinskii, Seppo Hassi, $Q$-functions and boundary triplets of nonnegative operators, in: Recent advances in inverse scattering, Schur analysis and stochastic processes, Birkhauser/Springer, Cham, 2015.  MathSciNet
10. Yu. M. Arlinskii, S. Hassi, H. S. V. Snoo de, $Q$-functions of Hermitian contractions of Krei n-Ovcharenko type, Integral Equations Operator Theory 53 (2005), no. 2, 153-189.  MathSciNet CrossRef
11. Yury Arlinskii, Seppo Hassi, Henk Snoo de, $Q$-functions of quasi-selfadjoint contractions, in: Operator theory and indefinite inner product spaces, Birkhauser, Basel, 2006.  MathSciNet CrossRef
12. Yury M. Arlinskii, Seppo Hassi, Henk S. V. Snoo de, Parametrization of contractive block operator matrices and passive discrete-time systems, Complex Anal. Oper. Theory 1 (2007), no. 2, 211-233.  MathSciNet CrossRef
13. Yury M. Arlinskii, Seppo Hassi, Henk S. V. Snoo de, Passive systems with a normal main operator and quasi-selfadjoint systems, Complex Anal. Oper. Theory 3 (2009), no. 1, 19-56.  MathSciNet CrossRef
14. Yu. Arlinskii and E. Tsekanovskii, Non-self-adjoint contractive extensions of a Hermitian contraction and theorem of M. G. Krein, Uspekhi Mat. Nauk 37 (1982), no. 1, 131-132. (Russian); English transl. Russian Math. Surveys 37 (1982), no. 1, 151-152.
15. Yu. Arlinskii and E. Tsekanovskii, Quasi-self-adjoint contractive extensions of a Hermitian contraction, Teor. Funktsii, Funktsional. Anal. i Prilozhen. 50 (1988), 9-16. (Russian); English transl. J. Soviet Math. 49 (1990), no. 6, 1241-1247. CrossRef
16. D. Z. Arov, Passive linear steady-state dynamical systems, Sibirsk. Mat. Zh. 20 (1979), no. 2, 211-228, 457.  MathSciNet
17. Earl A. Coddington, Selfadjoint subspace extensions of nondensely defined symmetric operators, Bull. Amer. Math. Soc. 79 (1973), 712-715.  MathSciNet
18. Earl A. Coddington, Extension theory of formally normal and symmetric subspaces, American Mathematical Society, Providence, R.I., 1973.  MathSciNet
19. V. A. Derkach, M. M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 (1991), no. 1, 1-95.  MathSciNet CrossRef
20. V. A. Derkach, M. M. Malamud, The extension theory of Hermitian operators and the moment problem, J. Math. Sci. 73 (1995), no. 2, 141-242.  MathSciNet CrossRef
21. Vladimir Derkach, Seppo Hassi, Mark Malamud, Henk Snoo de, Boundary relations and their Weyl families, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5351-5400.  MathSciNet CrossRef
22. A. Dijksma, H. S. V. Snoo de, Self-adjoint extensions of symmetric subspaces, Pacific J. Math. 54 (1974), 71-100.  MathSciNet
23. R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413-415.  MathSciNet
24. P. A. Fillmore, J. P. Williams, On operator ranges, Advances in Math. 7 (1971), 254-281.  MathSciNet
25. Seppo Hassi, Mark Malamud, Henk Snoo de, On Krei ns extension theory of nonnegative operators, Math. Nachr. 274/275 (2004), 40-73.  MathSciNet CrossRef
26. Tosio Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, 1995.  MathSciNet
27. M. G. Krein, On Hermitian operators with defect indices equal to Unity, Dokl. Akad. Nauk SSSR 43 (1944), no. 8, 339-342. (Russian)
28. M. G. Krein, Resolvents of an Hermitian operator with defect index $(m, m)$, Dokl. Akad. Nauk SSSR 52 (1946), 657-660. (Russian)
29. M. G. Krein, Theory of selfadjoint extensions of semibounded operators and its applications. I, Mat. Sb. 20 (1947), no. 3, 431-498. (Russian)
30. M. G. Krein, The description of all solutions of the truncated power moment problem and some problems of operator theory, Mat. Issled. 2 (1967), no. vyp. 2, 114-132.  MathSciNet
31. M. G. Krein, G. K. Langer, The defect subspaces and generalized resolvents of a Hermitian operator in the space $\Pi _\kappa $, Funkcional. Anal. i Prilo\v zen 5 (1971), no. 2, 59-71.  MathSciNet
32. M. G. Krein, G. K. Langer, The defect subspaces and generalized resolvents of a Hermitian operator in the space $\Pi _\kappa $, Funkcional. Anal. i Prilo\v zen 5 (1971), no. 3, 54-69.  MathSciNet
33. M. G. Krein, I. E. Ovcarenko, $Q$-functions and $sc$-resolvents of nondensely defined Hermitian contractions, Sibirsk. Mat. \v Z. 18 (1977), no. 5, 1032-1056, 1206.  MathSciNet
34. H. Langer, B. Textorius, On generalized resolvents and $Q$-functions of symmetric linear relations (subspaces) in Hilbert space, Pacific J. Math. 72 (1977), no. 1, 135-165.  MathSciNet
35. Heinz Langer, Bjorn Textorius, Generalized resolvents of dual pairs of contractions, in: Invariant subspaces and other topics (Timi\c soara/Herculane, 1981), Birkhauser, Basel-Boston, Mass., 1982.  MathSciNet
36. M. Neumark, Self-adjoint extensions of the second kind of a symmetric operator, Bull. Acad. Sci. URSS. S\er. Math. [Izvesti\`a Akad. Nauk SSSR] 4 (1940), 53-104.  MathSciNet
37. M. Neumark, Spectral functions of a symmetric operator, Bull. Acad. Sci. URSS. S\er. Math. [Izvestia Akad. Nauk SSSR] 4 (1940), 277-318.  MathSciNet
38. M. A. Neumark, On spectral functions of a symmetric operator, Bull. Acad. Sci. URSS. S\er. Math. [Izvestia Akad. Nauk SSSR] 7 (1943), 285-296.  MathSciNet
39. F. S. Rofe-Beketov, The numerical range of a linear relation and maximum relations, Teor. Funktsi\u\i\ Funktsional. Anal. i Prilozhen. (1985), no. 44, 103-112.  MathSciNet CrossRef
40. A. V. Straus, Generalized resolvents of symmetric operators, Izvestiya Akad. Nauk SSSR. Ser. Mat. 18 (1954), 51-86.  MathSciNet
41. Konrad Schmudgen, On domains of powers of closed symmetric operators, J. Operator Theory 9 (1983), no. 1, 53-75.  MathSciNet
42. Yu. L. Shmulyan, An operator Hellinger integral, Mat. Sb. (N.S.) 49 (1959), no. 4, 381-430. (Russian)
43. Ju. L. Smul′jan, Certain stability properties for analytic operator-valued functions, Mat. Zametki 20 (1976), no. 4, 511-520.  MathSciNet
44. Bela Sz.-Nagy, Ciprian Foia\lfhooks, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akad\'emiai Kiad\'o, Budapest, 1970.  MathSciNet
45. Sergey M. Zagorodnyuk, Generalized resolvents of symmetric and isometric operators: the Shtraus approach, Ann. Funct. Anal. 4 (2013), no. 1, 175-285.  MathSciNet CrossRef