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Transactions of the American Mathematical Society

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Complex symmetric operators and applications

Authors: Stephan Ramon Garcia and Mihai Putinar
Journal: Trans. Amer. Math. Soc. 358 (2006), 1285-1315
MSC (2000): Primary 30D55, 47A15
Published electronically: May 26, 2005
MathSciNet review: 2187654
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Abstract: We study a few classes of Hilbert space operators whose matrix representations are complex symmetric with respect to a preferred orthonormal basis. The existence of this additional symmetry has notable implications and, in particular, it explains from a unifying point of view some classical results. We explore applications of this symmetry to Jordan canonical models, self-adjoint extensions of symmetric operators, rank-one unitary perturbations of the compressed shift, Darlington synthesis and matrix-valued inner functions, and free bounded analytic interpolation in the disk.

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  • 1. N. I. Akhiezer, The classical moment problem and some related questions in analysis, Translated by N. Kemmer, Hafner Publishing Co., New York, 1965. MR 0184042
  • 2. Daniel Alpay, Aad Dijksma, James Rovnyak, and Hendrik de Snoo, Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Operator Theory: Advances and Applications, vol. 96, Birkhäuser Verlag, Basel, 1997. MR 1465432
  • 3. P. R. Ahern and D. N. Clark, On inner functions with 𝐻^{𝑝}-derivative, Michigan Math. J. 21 (1974), 115–127. MR 0344479
  • 4. D. Z. Arov, Darlington’s method in the study of dissipative systems, Dokl. Akad. Nauk SSSR 201 (1971), no. 3, 559–562 (Russian); Russian transl., Soviet Physics Dokl. 16 (1971), 954–956 (1972). MR 0428098
  • 5. D. Z. Arov, Stable dissipative linear stationary dynamical scattering systems, Interpolation theory, systems theory and related topics (Tel Aviv/Rehovot, 1999) Oper. Theory Adv. Appl., vol. 134, Birkhäuser, Basel, 2002, pp. 99–136. Translated from the Russian [J. Operator Theory 2 (1979), no. 1, 95–126; MR0553866] and with appendices by the author and J. Rovnyak. MR 2013544
  • 6. D. Z. Arov, Realization of matrix-valued functions according to Darlington, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 1299–1331 (Russian). MR 0357820
  • 7. Š. Asadi and I. E. Lucenko, Antiunitary transformations of linear operators, Vestnik Har′kov. Gos. Univ. 83 Mat. i Meh. Vyp. 37 (1972), 13–20, 120 (Russian). MR 0333778
  • 8. Hari Bercovici, Operator theory and arithmetic in 𝐻^{∞}, Mathematical Surveys and Monographs, vol. 26, American Mathematical Society, Providence, RI, 1988. MR 954383
  • 9. S. Bergman and M. Schiffer, Kernel functions and conformal mapping, Compositio Math. 8 (1951), 205–249. MR 0039812
  • 10. Joseph A. Cima and William T. Ross, The backward shift on the Hardy space, Mathematical Surveys and Monographs, vol. 79, American Mathematical Society, Providence, RI, 2000. MR 1761913
  • 11. Douglas N. Clark, One dimensional perturbations of restricted shifts, J. Analyse Math. 25 (1972), 169–191. MR 0301534
  • 12. deWilde, P., Roomy scattering matrix synthesis, Technical Report (Berkeley, 1971).
  • 13. R. G. Douglas and J. William Helton, Inner dilations of analytic matrix functions and Darlington synthesis, Acta Sci. Math. (Szeged) 34 (1973), 61–67. MR 0322538
  • 14. R. G. Douglas, H. S. Shapiro, and A. L. Shields, Cyclic vectors and invariant subspaces for the backward shift operator., Ann. Inst. Fourier (Grenoble) 20 (1970), no. fasc. 1, 37–76 (English, with French summary). MR 0270196
  • 15. Peter L. Duren, Theory of 𝐻^{𝑝} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
  • 16. Ciprian Foias and Arthur E. Frazho, The commutant lifting approach to interpolation problems, Operator Theory: Advances and Applications, vol. 44, Birkhäuser Verlag, Basel, 1990. MR 1120546
  • 17. Kurt Friedrichs, On certain inequalities and characteristic value problems for analytic functions and for functions of two variables, Trans. Amer. Math. Soc. 41 (1937), no. 3, 321–364. MR 1501907, 10.1090/S0002-9947-1937-1501907-0
  • 18. F. R. Gantmacher, The theory of matrices. Vols. 1, 2, Translated by K. A. Hirsch, Chelsea Publishing Co., New York, 1959. MR 0107649
  • 19. F. P. Gantmacher and M. G. Krein, Oscillation matrices and kernels and small vibrations of mechanical systems, Revised edition, AMS Chelsea Publishing, Providence, RI, 2002. Translation based on the 1941 Russian original; Edited and with a preface by Alex Eremenko. MR 1908601
  • 20. Garcia, S.R., A $^*$-closed subalgebra of the Smirnov class, Proc. Amer. Math. Soc. 133 (2005), 2051-2059.
  • 21. Garcia, S.R., Conjugation, the backward shift, and Toeplitz kernels, to appear: Journal of Operator Theory.
  • 22. Garcia, S.R., Inner matrices and Darlington synthesis, to appear: Methods Funct. Anal. Topology.
  • 23. Stephan Ramon Garcia and Donald Sarason, Real outer functions, Indiana Univ. Math. J. 52 (2003), no. 6, 1397–1412. MR 2021044, 10.1512/iumj.2003.52.2511
  • 24. I. M. Glazman, Pryamye metody kachestvennogo spektralnogo analiza, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR 0185471
  • 25. V. Ī. Godič and I. E. Lucenko, The structure of bisymmetric operators, Teor. Funkciĭ Funkcional. Anal. i Priložen. 16 (1972), 138–139, 219 (Russian). MR 0344934
  • 26. I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. MR 0246142
  • 27. I. C. Gohberg and M. G. Kreĭn, Theory and applications of Volterra operators in Hilbert space, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 24, American Mathematical Society, Providence, R.I., 1970. MR 0264447
  • 28. M. L. Gorbachuk and V. I. Gorbachuk, M. G. Krein’s lectures on entire operators, Operator Theory: Advances and Applications, vol. 97, Birkhäuser Verlag, Basel, 1997. MR 1466698
  • 29. Hans Ludwig Hamburger, Über die Zerlegung des Hilbertschen Raumes durch vollstetige lineare Transformationen, Math. Nachr. 4 (1951), 56–69 (German). MR 0040587
  • 30. Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183
  • 31. Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
  • 32. M. G. Kreĭn and M. A. Naĭmark, The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations, Linear and Multilinear Algebra 10 (1981), no. 4, 265–308. Translated from the Russian by O. Boshko and J. L. Howland. MR 638124, 10.1080/03081088108817420
  • 33. Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
  • 34. Henry Helson, Large analytic functions. II, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 217–220. MR 1044789
  • 35. B. A. Lotto and D. Sarason, Multiplicative structure of de Branges’s spaces, Rev. Mat. Iberoamericana 7 (1991), no. 2, 183–220. MR 1133377, 10.4171/RMI/110
  • 36. I. E. Lucenko, Linear operators that commute with antiunitary operators, Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp. 9 (1969), 85–93 (Russian). MR 0278085
  • 37. von Neumann, J., Allgemeine Eigenwerttheorie Hermitischer Funktionaloperatoren, Math. Ann. 102 (1929), 49-131.
  • 38. N. K. Nikol′skiĭ, Treatise on the shift operator, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin, 1986. Spectral function theory; With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller; Translated from the Russian by Jaak Peetre. MR 827223
  • 39. Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 2, Mathematical Surveys and Monographs, vol. 93, American Mathematical Society, Providence, RI, 2002. Model operators and systems; Translated from the French by Andreas Hartmann and revised by the author. MR 1892647
  • 40. Vladimir V. Peller, Hankel operators and their applications, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. MR 1949210
  • 41. V. P. Potapov, The multiplicative structure of 𝐽-contractive matrix functions, Trudy Moskov. Mat. Obšč. 4 (1955), 125–236 (Russian). MR 0076882
  • 42. Mihai Putinar and Harold S. Shapiro, The Friedrichs operator of a planar domain, Complex analysis, operators, and related topics, Oper. Theory Adv. Appl., vol. 113, Birkhäuser, Basel, 2000, pp. 303–330. MR 1771771
    Mihai Putinar and Harold S. Shapiro, The Friedrichs operator of a planar domain. II, Recent advances in operator theory and related topics (Szeged, 1999), Oper. Theory Adv. Appl., vol. 127, Birkhäuser, Basel, 2001, pp. 519–551. MR 1902820
  • 43. Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
  • 44. Stefan Richter and Carl Sundberg, A formula for the local Dirichlet integral, Michigan Math. J. 38 (1991), no. 3, 355–379. MR 1116495, 10.1307/mmj/1029004388
  • 45. Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, 1990. Translated from the second French edition by Leo F. Boron; Reprint of the 1955 original. MR 1068530
  • 46. William T. Ross and Harold S. Shapiro, Generalized analytic continuation, University Lecture Series, vol. 25, American Mathematical Society, Providence, RI, 2002. MR 1895624
  • 47. Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841
  • 48. Menahem Schiffer, Fredholm eigenvalues and Grunsky matrices, Ann. Polon. Math. 39 (1981), 149–164. MR 617457
  • 49. Carl Ludwig Siegel, Symplectic geometry, Amer. J. Math. 65 (1943), 1–86. MR 0008094
  • 50. Barry Simon, Spectral analysis of rank one perturbations and applications, Mathematical quantum theory. II. Schrödinger operators (Vancouver, BC, 1993) CRM Proc. Lecture Notes, vol. 8, Amer. Math. Soc., Providence, RI, 1995, pp. 109–149. MR 1332038
  • 51. Takagi, T., On an algebraic problem related to an analytic theorem of Caratheodory and Fejer and on an allied theorem of Landau, Japan J. Math. 1 (1925), 83-93.
  • 52. Olga Taussky, The role of symmetric matrices in the study of general matrices, Linear Algebra and Appl. 5 (1972), 147–154. MR 0302674
  • 53. James E. Thomson, Approximation in the mean by polynomials, Ann. of Math. (2) 133 (1991), no. 3, 477–507. MR 1109351, 10.2307/2944317

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Additional Information

Stephan Ramon Garcia
Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106-3080

Mihai Putinar
Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106-3080

Keywords: Complex symmetric operators, interpolation, self-adjoint extension, Takagi factorization, shift operators, inner functions, Darlington synthesis, Clark perturbations, Jordan operators, Volterra operators
Received by editor(s): February 21, 2004
Received by editor(s) in revised form: May 10, 2004
Published electronically: May 26, 2005
Additional Notes: The second author was supported in part by NSF Grant DMS #0100367.
Article copyright: © Copyright 2005 American Mathematical Society