Skip to main content
SpringerLink
Log in

Accretive (*)-extensions and Realization Problems

  • Conference paper
  • First Online:
Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 221))

  • 1089 Accesses

Abstract

We present a solution of the extended Phillips-Kato extension problem about existence and parametrization of all accretive (*)-extensions (with the exit into triplets of rigged Hilbert spaces) of a densely defined non-negative operator.I n particular, the analogs of the von Neumann and Friedrichs theorems for existence of non-negative self-adjoint (*)-extensions are obtained. Relying on these results we introduce the extremal classes of Stieltjes and inverse Stieltjes functions and show that each function from these classes can be realized as the impedance function of an L-system.I t is proved that in this case the realizing L-system contains an accretive operator and, in case of Stieltjes functions, an accretive (*)-extension.M oreover, we establish the connection between the above-mentioned classes and the Friedrichs and Kre”n-von Neumann extremal non-negative extensions.

Mathematics Subject Classification (2000). Primary 47A10, 47B44; Secondary 46E20, 46F05.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Dover, New York, (1993)

    MATH  Google Scholar 

  2. Ando, T., Nishio, K.: Positive Selfadjoint Extensions of Positive Symmetric Operators. Tohóku Math. J., 22, 65-75 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arlinskii, Yu.M.: On inverse problem of the theory of characteristic functions of unbounded operator colligations. Dopovidi Akad. Nauk Ukrain. RSR, Ser. A, No. 2, 105-109 (1976)

    MathSciNet  Google Scholar 

  4. Arlinskii, Yu.M.: Regular (*)-extensions of quasi-Hermitian operators in rigged Hilbert spaces. (Russian), Izv. Akad. Nauk Armyan. SSR, Ser.Mat., 14, No. 4, 297-312 (1979)

    MathSciNet  Google Scholar 

  5. Arlinskii, Yu.M.: On regular (*)-extensions and characteristic matrix-valued functions of ordinary differential operators. Boundary value problems for differential operators, Kiev, 3-13, (1980)

    Google Scholar 

  6. Arlinskii, Yu.M: On accretive (*)-extensions of a positive symmetric operator. (Russian), Dokl. Akad. Nauk. Ukraine, Ser. A, No. 11, 3-5 (1980)

    Google Scholar 

  7. Arlinskii, Yu.M., Tsekanovskii, E.R.: Nonselfadjoint contractive extensions of a Her-mitian contraction and theorems of Krein. Russ. Math. Surv., 37:1, 151-152 (1982)

    Article  MathSciNet  Google Scholar 

  8. Arlinskii, Yu.M., Tsekanovskii, E.R.: Sectorial extensions of positive Hermitian operators and their resolvents. (Russian), Akad. Nauk. Armyan. SSR, Dokl., 79, No. 5, 199-203 (1984)

    MathSciNet  Google Scholar 

  9. Arlinskii, Yu.M., Tsekanovskii, E.R.: Regular (*)-extension of unbounded operators, characteristic operator-functions and realization problems of transfer mappings of linear systems. Preprint, VINITI, 2867-79, Dep.-72 (1979)

    Google Scholar 

  10. Yu.M. Arlinskiiand E.R. Tsekanovskii: Quasi-self-adjoint contractive extensions of Hermitian contractions, Teor. Funkts., Funkts. Anal. Prilozhen, 50, 9-16 (1988) (Russian). English translation in J. Math. Sci. 49, No. 6, 1241-1247 (1990).

    Google Scholar 

  11. Arlinskii, Yu.M., Tsekanovskii, E.R: Linear systems with Schrödinger operators and their transfer functions. Oper. Theory Adv. Appl., 149, 47-77 (2004)

    MathSciNet  Google Scholar 

  12. Arlinskii, Yu.M., Tsekanovskii, E.R: The von Neumann problem for nonnegative symmetric operators. Integral Equations Operator Theory, 51, No. 3, 319-356 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Belyi, S.V., Hassi, S., de Snoo, H.S.V., Tsekanovskii, E.R.: On the realization of inverse of Stieltjes functions. Proceedings of MTNS-2002, University of Notre Dame, CD-ROM, 11p., (2002)

    Google Scholar 

  14. Belyi, S.V., Tsekanovskii, E.R.: Realization theorems for operator-valued R-functions. Operator theory: Advances and Applications, 98,Birkhäuser Verlag Basel, 55-91 (1997)

    Google Scholar 

  15. Belyi, S.V., Tsekanovskii, E.R.: Multiplication Theorems for J-contractive operator-valued functions. Fields Institute Communications, 25, 187-210 (2000)

    MathSciNet  Google Scholar 

  16. Belyi, S.V., Tsekanovskii, E.R.: On classes of realizable operator-valued R-functions. Operator theory: Advances and Applications, 115, Birkhäuser Verlag Basel, (2000), 85-112.

    Google Scholar 

  17. Belyi, S.V., Tsekanovskii, E.R.: Stieltjes like functions and inverse problems for systems with Schrödinger operator. Operators and Matrices, vol. 2, No. 2, 265-296 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Belyi, S.V., Tsekanovskii, E.R.: Inverse Stieltjes like functions and inverse problems for systems with Schrödinger operator. Operator Theory: Advances and Applications, vol. 197, 21-49 (2009)

    MathSciNet  Google Scholar 

  19. Douglas, R.G.: On majorization, factorization and range inclusion of operators in Hilbert space. Proc. Amer. Math. Soc. 17, 413-416 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dovzhenko, I., Tsekanovskii, E.R.: Classes of Stieltjes operator-valued functions and their conservative realizations. Soviet Math. Dokl., 41, no. 2, 201-204 (1990)

    MathSciNet  MATH  Google Scholar 

  21. Fillmore, P.A., Williams, J.P.: On operator ranges. Advances in Math. 7, 254-281 (1971)

    MathSciNet  MATH  Google Scholar 

  22. Gesztesy, F., Tsekanovskii, E.R.: On Matrix-Valued Herglotz Functions. Math. Nachr. 218, 61-138 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kac, I.S., Krein, M.G.:.R-functions - analytic functions mapping the upper half-plane into itself. Amer. Math. Soc. Transl., (2), 103, 1-18 (1974)

    MATH  Google Scholar 

  24. Kato, T.: Perturbation Theory for Linear Operators. Springer-Verlag, (1966)

    Google Scholar 

  25. Krein, M.G.: The Theory of Selfadjoint Extensions of Semibounded Hermitian Transformations and its Applications. I, (Russian) Mat. Sbornik 20, No. 3, 431-495 (1947)

    MathSciNet  Google Scholar 

  26. Krein, M.G.: The Theory of Selfadjoint Extensions of Semibounded Hermitian Transformations and its Applications, II, (Russian) Mat. Sbornik 21, No. 3, 365-404 (1947)

    MathSciNet  Google Scholar 

  27. Krein, M.G., Langer, H.: Über die Q-Function eines II-Hermiteschen Operators im Raum II. Acta Sci. Math. Szeged, 34, 191-230 (1973)

    MathSciNet  MATH  Google Scholar 

  28. Malamud, M.: On some classes of Hermitian operators with gaps. (Russian) Ukrainian Mat.J., 44, No. 2, 215-234 (1992)

    Article  MathSciNet  Google Scholar 

  29. Naimark, M.A.: Linear Differential Operators II. F. Ungar Publ., New York, (1968)

    MATH  Google Scholar 

  30. Okunskii, M.D., Tsekanovskii, E.R.: On the theory of generalized selfadjoint extensions of semibounded operators. (Russian) Funkcional. Anal. i Prilozen., 7, No. 3, 92-93 (1973)

    MathSciNet  Google Scholar 

  31. Phillips, R.: On dissipative operators, in "Lectures in Differential Equations", vol. II, Van Nostrand-Reinhold, New York, 65-113 (1965).

    Google Scholar 

  32. Tsekanovskiii, E.R.: Non-self-adjoint accretive extensions of positive operators and theorems of Friedrichs-Krein-Phillips. Funct. Anal. Appl. 14, 156-157 (1980)

    Article  Google Scholar 

  33. Tsekanovskii, E.R.: Friedrichs and Krein extensions of positive operators and holo-morphic contraction semigroups. Funct. Anal. Appl. 15, 308-309 (1981)

    Article  MathSciNet  Google Scholar 

  34. Tsekanovskii, E.R.: Accretive Extensions and Problems on Stieltjes Operator-Valued Functions Relations. Operator Theory: Advan. and Appl., 59, 328-347 (1992)

    MathSciNet  Google Scholar 

  35. Tsekanovskii, E.R., Smuljan, Yu.L.: The theory of bi-extensions of operators on rigged Hilbert spaces. Unbounded operator colligations and characteristic functions. Russ. Math. Surv., 32, 73-131 (1977)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, East Ukrainian National University, Kvartal Molodyozhny, 20-A, 91034, Lugansk, Ukraine

    Yury Arlinskiĭ

  2. Department of Mathematics, Troy University, Troy, AL, 36082, USA

    Sergey Belyi

  3. Department of Mathematics, Niagara University, New York, 14109, USA

    Eduard Tsekanovskiĭ

Authors
  1. Yury Arlinskiĭ
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergey Belyi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eduard Tsekanovskiĭ
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yury Arlinskiĭ .

Editor information

Editors and Affiliations

  1. , Abt. Mathematik V, Universität Ulm, Ulm, 89069, Germany

    Wolfgang Arendt

  2. Department of Mathematics, Virginia Polytechnic Institute, McBryde Hall 524, Blacksburg, 24061, Virginia, USA

    Joseph A. Ball

  3. Institut für Mathematik, AG Modellierung, Numerik und, TU Berlin, Straße des 17 Juni 136, Berlin, 10623, Berlin, Germany

    Jussi Behrndt

  4. Fak. II Mathematik & Naturwissenschaften, Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Berlin, Germany

    Karl-Heinz Förster

  5. Fak. II Mathematik & Naturwissenschaften, Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Berlin, Germany

    Volker Mehrmann

  6. , Institut für Mathematik, TU Ilmenau, Weimarer Str. 25, Ilmenau, 98693, Thüringen, Germany

    Carsten Trunk

Additional information

Dedicated to Heinz Langer on the occasion of his 75th birthday

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Basel

About this paper

Cite this paper

Arlinskiĭ, Y., Belyi, S., Tsekanovskiĭ, E. (2012). Accretive (*)-extensions and Realization Problems. In: Arendt, W., Ball, J., Behrndt, J., Förster, KH., Mehrmann, V., Trunk, C. (eds) Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations. Operator Theory: Advances and Applications, vol 221. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0297-0_5

Download citation

Publish with us

Policies and ethics

Search

Navigation