Skip to main content
SpringerLink
Log in

Extension Theorems for Contraction Operators on Kreĭn Spaces

  • Chapter
Extension and Interpolation of Linear Operators and Matrix Functions

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 47))

Abstract

Notions of Julia and defect operators are used as a foundation for a theory of matrix extension and commutant lifting problems for contraction operators on Kreĭn spaces. The account includes a self-contained treatment of key propositions from the theory of Potapov, Ginsburg, Kreĭn, and Shmul’yan on the behavior of a contraction operator on negative subspaces. This theory is extended by an analysis of the behavior of the adjoint of a contraction operator on negative subspaces. Together, these results provide the technical input for the main extension theorems.

The second author was supported by the National Science Foundation.

To the memory of Mark Grigor’eviĭ Kreĭn

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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. D. Alpay, Dilatations des commutants d’operateurs pour des espaces de Krein de fonctions analytiques,Annales de l’Institut Fourier, to appear.

    Google Scholar 

  2. D. Alpay, Some remarks on reproducing kernel Krein spaces,Rocky Mountain J. Math., to appear.

    Google Scholar 

  3. D. Alpay, The Schur Algorithm, System Theory and Reproducing Kernel Hilbert Spaces of Analytic Functions, lecture notes from Groningen University, 1989.

    Google Scholar 

  4. T. Andó, Linear Operators on Krein Spaces, Hokkaido University, Research Institute of Applied Electricity, Division of Applied Mathematics, Sapporo, 1979. MR 81e: 47032.

    Google Scholar 

  5. D. Z. Arov, Darlington realization of matrix-valued functions,Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 1299-1331; Math. USSR-Izv. 7 (1973), 1295-1326. MR 50#10287.

    Google Scholar 

  6. Gr. Arsene, Z. Ceau§escu, and C. Foial, On intertwining dilations. VIII, J. Operator Theory 4 (1980), 55-91. MR 82d: 47013f.

    Google Scholar 

  7. Gr. Arsene and A. Ghéondea, Completing matrix contractions, J. Operator Theory 7 (1982), 179-189. MR 83i: 4710.

    Google Scholar 

  8. Gr. Arsene, T. Constantinescu, and A. Ghéondea, Lifting of operators and prescribed numbers of negative squares, Michigan J. Math. 34 (1987), 201-216. MR 88j: 47008.

    Google Scholar 

  9. T. Azizov and I. S. Iokhvidov, Linear operators in spaces with an indefinite metric and their applications, Mathematical Analysis, Vol. 17, pp. 113-205, 272, Akad. Nauk SSSR, Vsesojuz. Inst. Nauen. i Tehn. Informacii, Moscow, 1979; J. Soviet Math. 15 (1981), 438-490. MR 81m: 47052.

    Google Scholar 

  10. T. Azizov and I. S. Iokhvidov, Foundations of the Theory of Linear Operators in Spaces with Indefinite Metric, “Nauka”, Moscow, 1986; English transi. Linear Operators in Spaces with Indefinite Metric, Wiley, New York, 1989. MR 88g: 47070.

    Google Scholar 

  11. J. A. Ball and J. W. Helton, A Beurling-Lax theorem for the Lie group U(m, n) which contains most interpolation theory, J. Operator Theory 9 (1983), 107142. MR 84m: 47046.

    Google Scholar 

  12. J. Bognâr, Indefinite Inner Product Spaces,Springer, Berlin-New York, 1974. MR 57#7125.

    Google Scholar 

  13. L. de Branges, Complementation in Krein spaces,Trans. Amer. Math. Soc. 305 (1988), 277-291. MR 89c:46034.

    Google Scholar 

  14. L. de Branges, Krein spaces of analytic functions,J. Funct. Anal. 81 (1988), 219-259.

    Google Scholar 

  15. L. de Branges, Square Summable Power Series,in preparation.

    Google Scholar 

  16. L. de Branges, A construction of Krein spaces of analytic functions,preprint, 1989.

    Google Scholar 

  17. L. de Branges, Commutant lifting in Krein spaces,in preparation.

    Google Scholar 

  18. L. de Branges and J. Rovnyak, Canonical models in quantum scattering theory,Perturbation Theory and its Applications in Quantum Mechanics, (Proc. Adv. Sem. Math. Res. Center, U. S. Army, Theoret. Chem. Inst., Univ. of Wisconsin, Madison, Wis., 1965), C. H. Wilcox, ed., pp. 295-392, Wiley, New York, 1966. MR 39#6109.

    Google Scholar 

  19. L. de Branges and J. Rovnyak, Square Summable Power Series,Holt, Rinehart and Winston, New York, 1966. MR 35#5909.

    Google Scholar 

  20. P. Bruinsma, A. Dijksma, H. de Snoo, Unitary dilations of contractions in II, spaces, Oper. Theory: Adv. Appl., Vol. 28, pp. 27-42, Birkhäuser, Basel-Boston, 1988. MR 90b: 47062.

    Google Scholar 

  21. T. Constantinescu and A. Ghéondea, On unitary dilations and characteristic functions in indefinite product spaces, Oper. Theory: Adv. Appl., Vol. 24, pp. 87-102, Birkhäuser, Basel-Boston, 1987. MR 88j: 47049.

    Google Scholar 

  22. T. Constantinescu and A. Ghéondea, Minimal signature in lifting of operators. I,II, INCREST, Bucharest, preprints; Part I: No. 56/1988; Part I I: March 1989.

    Google Scholar 

  23. Ch. Davis, J-unitary dilation of a general operator Acta Sci. Math. (Szeged) 31 (1970), 75-86. MR 41#9032.

    Google Scholar 

  24. Ch. Davis, An extremal problem for extensions of a sesquilinear form Linear Algebra and Appl. 13 (1976), 91-102. MR 52#15068.

    Google Scholar 

  25. Ch. Davis, A factorization of an arbitrary m x n contractive operator-matrix, Toeplitz Centennial (Tel Aviv, 1981), Oper. Theory: Adv. Appl., Vol. 4, pp. 217-232, Birkhäuser, Basel-Boston, 1982. MR 84c: 47011

    Google Scholar 

  26. Ch. Davis, W. M. Kahan, and H. F. Weinberger, Norm-preserving dilations and their applications to optimal error bounds, SIAM J. Numer. Anal. 19 (1982), 445-469. MR 84b: 47010.

    Google Scholar 

  27. J. Dieudonné, Quasi-hermitian operators Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), pp. 115-122, Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961. MR 32#4540.

    Google Scholar 

  28. R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert spaces Proc. Amer. Math. Soc. 17 (1966), 413-415. MR 34#3315.

    Google Scholar 

  29. M. Dritschel, A lifting theorem for bicontractions, J. Funct. Anal. 88 (1990), 61 - 89.

    Google Scholar 

  30. M. Dritschel, Extension Theorems for Operators on Kreän Spaces, Dissertation, University of Virginia, 1989.

    Google Scholar 

  31. N. Dunford and J. T. Schwartz, Linear Operators Vol. I, Interscience, New York, 1958. MR22#8302.

    Google Scholar 

  32. H. Dym, J Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces, and Interpolation, CBMS Regional Conference Series in Mathematics, Vol. 71, Amer. Math. Soc., Providence, 1989.

    Google Scholar 

  33. C. Foim, Contractive intertwining dilations and waves in layered media, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 605-613, Acad. Sci. Fennica, Helsinki, 1980. MR 82a: 47002.

    Google Scholar 

  34. C. Foiaq and A. Frazho, On the Schur representation in the commutant lifting problem. 1,II, Part I: I. Schur Methods in Operator Theory and Signal Processing, Oper. Theory: Adv. Appl., Vol. 18, pp. 207-217, Birkhäuser, Basel-Boston, 1986; Part II: Oper. Theory: Adv. Appl., Vol. 29, pp. 171-179, Birkhäuser, Basel-Boston, 1988. MR 89a: 47011.

    Google Scholar 

  35. A. Frazho, Three inverse scattering algorithms for the lifting theorem, I. Schur Methods in Operator Theory and Signal Processing, Oper. Theory: Adv. Appl., Vol. 18, pp. 219-248, Birkhäuser, Basel-Boston, 1986. MR 89c: 47012.

    Google Scholar 

  36. A. Ghéondea, Canonical forms of unbounded unitary operators in Krein spaces, Publ. Res. Inst. Math. Sci. 24 (1988), no. 2, 205-224. MR 90a: 47087.

    Google Scholar 

  37. Yu. P. Ginsburg, J-nonexpansive analytic operator-functions, Dissertation, Odessa, 1958.

    Google Scholar 

  38. I. Gohberg, P. Lancaster, and L. Rodman, Matrices and Indefinite Scalar Products, Birkhäuser, Basel-Boston, 1983. MR 87j: 15001.

    Google Scholar 

  39. J. W. Helton, with the assistance of J. A. Ball, C. R. Johnson, and J. N. Palmer, Operator Theory, Analytic Functions, Matrices, and Electrical Engioneering, CBMS Regional Conference Series in Mathematics, Vol. 68, Amer. Math. Soc., Providence, 1987. MR 89f: 47001.

    Google Scholar 

  40. I. S. Iokhvidov, M. G. Kreïn, and H. Langer, Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric, Akademie-Verlag, Berlin, 1982. MR 85g: 47050.

    Google Scholar 

  41. M. G. Krein, Completely continuous linear operators in function spaces with two norms, Akad. Nauk Ukrain. RSR. Zbirnik Prac’ Inst. Mat., No. 9, (1947), 104 - 129.

    Google Scholar 

  42. M. G. Kreïn, On an application of a fixed point principle in the theory of linear transformations of spaces with an indefinite metric, Uspekhi Mat. Nauk 5, No. 2, (1950), 180-190. MR 14, 56.

    Google Scholar 

  43. M. G. Krein, Introduction to the geometry of indefinite J-spaces and to the theory of operators in those spaces Second Math. Summer School, Part I, pp. 15-92. Naukova Dumka, Kiev, 1965; Amer. Math. Soc. Transl. (2) 93 (1970), 103-176. MR 33#574.

    Google Scholar 

  44. M. G. Kreïn and H. Langer, Continuation of Hermitian positive definite functions and related questions preprint.

    Google Scholar 

  45. M. G. Krein and Yu. L. Shmul’yan, Plus operators in a space with indefinite metric Mat. Issled. 1 (1966), 131-161; Amer. Math. Soc. Transi. (2) 85 (1969), 93-113. MR 34#4923.

    Google Scholar 

  46. M. G. Krein and Yu. L. Shmul’yan, J-polar representation of plus-operators Mat. Issled. 1 (1966), 172-210; Amer. Math. Soc. Transl. (2) 85 (1969), 115143. MR 34#8183.

    Google Scholar 

  47. M. G. Kreïn and Yu. L. Shmu’’yan, On linear-fractional transformations with operator coefficients Mat. Issled. 2 (1967), 64-96; Amer. Math. Soc. Transi. (2) 103 (1974), 125-152. MR 39#813.

    Google Scholar 

  48. S. A. Kuzhel’, J-nonexpansive operators, Teor. Funktsional. Anal. i. Prilozhen., No. 45, (1986), 63-68, ii. MR 87m: 47086.

    Google Scholar 

  49. H. Langer, Spectral functions of definitizable operators in Krein spaces, Functional analysis (Dubrovnik, 1981), pp. 1-46, Lecture Notes in Math., Vol. 948, Springer, Berlin-New York, 1982. MR 84g: 47034.

    Google Scholar 

  50. P. D. Lax, Symmetrizable linear transformations, Comm. Pure Appl. Math. 7 (1954), 633-647. MR 16, 832.

    Google Scholar 

  51. R. B. Leech, Factorization of analytic functions and operator inequalities unpublished manuscript (early 1970’s?).

    Google Scholar 

  52. B. McEnnis, Shifts on indefinite inner product spaces, Pacific J. Math. 81 (1979), 113-130; part II, ibid. 100 (1982), 177-183. MR 81c:47040; 84b: 47043.

    Google Scholar 

  53. B. McEnnis, Shifts on Krein Spaces„ Proceedings of Symposia in Pure Mathematics: Operator Theory, Operator Algebras, and Applications, Amer. Math. Soc., to appear.

    Google Scholar 

  54. S. Parrott, On the quotient norm and the Sz.-Nagy—Foia, lifting theorem, J. Funct. Anal. 30 (1978), 311-328. MR 81h: 47006.

    Google Scholar 

  55. V. P. Potapov, The multiplicative structure of J-contractive matrix functions, Trudy Moskov. Mat. Obshch. 4 (1955), 125-236; Amer. Math. Soc. Transi. (2) 15 (1960), 131-243. MR 17, 958.

    Google Scholar 

  56. V. Ptak and P. Vrbovâ, Lifting intertwining relations, Integral Equations and Operator Theory 11 (1988), 128 - 147.

    Article  Google Scholar 

  57. W. T. Reid, Symmetrizable completely continuous linear transformations in Hilbert space, Duke Math. J. 18 (1951), 41-56. MR 13, 564.

    Google Scholar 

  58. M. Rosenblum, A corona theorem for countably many functions, Integral Equations and Operator Theory 3 (1980), 125-137. MR 81e: 46034.

    Google Scholar 

  59. M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Oxford Math. Monographs, Oxford University Press, New York, 1985. MR 87e: 47001.

    Google Scholar 

  60. D. Sarason, Generalized interpolation in HO ° Trans. Amer. Math. Soc. 127 (1967), 179-203. MR 34#8193.

    Google Scholar 

  61. D. Sarason, Shift-invariant subspaces from the Brangesian point of view, The Bieberbach Conjecture (West Lafayette, Ind., 1985), 153-166, Math. Surveys Monographs, Vol. 21, Amer. Math. Soc., Providence, 1986. MR 88d: 47014a.

    Google Scholar 

  62. D. Sarason, Doubly shift-invariant spaces in H 2, J. Operator Theory 16 (1986), 75-97. MR 88d: 47014b.

    Google Scholar 

  63. L. Schwartz, Sous-espaces hilbertiens d’espaces vectoriels top_ ologiques et noyaux associés (noyaux reproduisants) J. Analyse Math. 13 (1964), 115-256. MR 31#3835.

    Google Scholar 

  64. Yu. L. Shmul’yan, Division in the class of J-expansive operators Mat. Sb. (N.S.) 74 (116) (1967), 516-525; Math. USSR-Sbornik 3 (1967), 471-479. MR 37#784.

    Google Scholar 

  65. Yu. L. Shmul’yan and R. N. Yanovskaya, Blocks of a contractive operator matrix, Izv. Vyssh. Uchebn. Zaved. Mat. 1981, No. 7, 72-75; Soviet Math. (Iz. VUZ) 25 (1981), No. 7, 82-86. MR 83e: 47007.

    Google Scholar 

  66. B. Sz.-Nagy and C. Foim, Harmonic Analysis of Operators on Hilbert Space North Holland, New York, 1970. MR 43#947.

    Google Scholar 

  67. Shao Zong Yan, The contraction operators on space II, Chinese Ann. Math. Ser. B 7 (1986), 75-89. MR 87m:47088.

    Google Scholar 

  68. Agnes Yang, A Construction of Krein spaces of Analytic Functions, Dissertation, Purdue University, 1990.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics Mathematical Sciences Building, Purdue University, West Lafayette, Indiana, 47907, USA

    Michael A. Dritschel

  2. Department of Mathematics Mathematics-Astronomy Building, University of Virginia, Charlottesville, Virginia, 22903, USA

    James Rovnyak

Authors
  1. Michael A. Dritschel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. James Rovnyak
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Raymond and Beverly Sackler Faculty of Exact Sciences, School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    I. Gohberg

Rights and permissions

Reprints and permissions

Copyright information

© 1990 Springer Basel AG

About this chapter

Cite this chapter

Dritschel, M.A., Rovnyak, J. (1990). Extension Theorems for Contraction Operators on Kreĭn Spaces. In: Gohberg, I. (eds) Extension and Interpolation of Linear Operators and Matrix Functions. Operator Theory: Advances and Applications, vol 47. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7701-5_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-7701-5_5

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-7643-2530-5

  • Online ISBN: 978-3-0348-7701-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation