Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-05-03742-6
Published electronically: May 26, 2005
MathSciNet review: 2187654
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. Akhiezer, N.I., The Classical Moment Problem, Oliver and Boyd, Edinburgh, 1965. MR 0184042 (32:1518)
  • 2. Alpay, D., Dijksma, A., Rovnyak, J., de Snoo, H., Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Birkhäuser, Basel, 1997. MR 1465432 (2000a:47024)
  • 3. Ahern, P., Clark, D., On inner functions with $H^p$ derivative, Michigan Math. J. 21 (1974), 115-127. MR 0344479 (49''9218)
  • 4. Arov, D.Z., Darlington's method in the study of dissipative systems, Dokl. Akad. Nauk SSSR 201 (1971), no. 3, 559-562. MR 0428098 (55:1127)
  • 5. Arov, D.Z., Stable dissipative linear stationary dynamical scattering systems, Operator Theory: Advances and Applications, 134 (2002), 99-136. MR 2013544 (2004j:47016)
  • 6. Arov, D.Z., Realization of matrix-valued functions according to Darlington, Math USSR Izvestija, 7(6) (1973), 1295-1326. MR 0357820 (50:10287)
  • 7. Asadi, S., Lucenko, I. E., Antiunitary transformations of linear operators (Russian), Vestnik Harkov. Gos. Univ. No. 83 Mat. i Meh. Vyp. 37 (1972), 13-20, 120. MR 0333778 (48:12102)
  • 8. Bercovici, H., Operator Theory and Arithmetic in $H^\infty$, Amer. Math. Soc., Providence, R.I., 1988. MR 0954383 (90e:47001)
  • 9. Bergman, S., Schiffer, M., Kernel functions and conformal mapping, Composition Math. 8 (1951), 205-249. MR 0039812 (12:602c)
  • 10. Cima, J.A., Ross, W.T., The Backward Shift on the Hardy Space, American Mathematical Society, Providence R.I., 2000. MR 1761913 (2002f:47068)
  • 11. Clark, D.N., One dimensional perturbations of restricted shifts, J. Anal. Math. 25, (1972) 169-191.MR R0301534 (46:692)
  • 12. deWilde, P., Roomy scattering matrix synthesis, Technical Report (Berkeley, 1971).
  • 13. Douglas, R.G., Helton, J. William, Inner dilations of analytic matrix functions and Darlington synthesis, Acta Sci. Math (Szeged), 34 (1973), 61-67. MR 0322538 (48:900)
  • 14. Douglas, R.G., Shapiro, H.S., Shields, A.L., Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier (Grenoble) 20 no. 1 (1970), 37-76. MR 0270196 (42:5088)
  • 15. Duren, P.L., Theory of $H^p$ Spaces, Pure and Appl. Math., Vol 38, Academic Press, New York, 1970.MR 0268655 (42:3552)
  • 16. Foias, C., Frazho, A.E., The commutant lifting approach to interpolation problems, Birkhäuser Verlag, Basel, 1990.MR 1120546 (92k:47033)
  • 17. Friedrichs, K., On certain inequalities for analytic functions and for functions of two variables, Trans. Amer. Math. Soc. 41 (1937), 321-364. MR 1501907
  • 18. Gantmacher, F.R., The theory of matrices, Volume II, Chelsea, New York, 1989. MR 0107649 (21:6372c)
  • 19. Gantmacher, F.R., Krein, M.G., Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems (Revised English Edition), American Math. Soc. Chelsea Publ., Amer. Math. Soc., Providence, R.I., 2002. MR 1908601 (2003f:34161)
  • 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. Garcia, S.R., Sarason, D., Real outer functions, Indiana Univ. Math. J., 52 (2003), 1397-1412. MR 2021044 (2004k:30129)
  • 24. Glazman, I.M., Direct methods of the qualitative spectral theory of singular differential operators (Russian), Gos. Iz. Fiz.-Mat. Lit., Moscow, 1963.MR 0185471 (32:2938)
  • 25. Godic, V.I., Lucenko, I.E., The structure of bisymmetric operators (Russian), Teoria Funkt. Funktional. Anal. Prilozh. 16 (1972), 138-139.MR 0344934 (49:9673)
  • 26. Gohberg, I., Krein, M.G., Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Amer. Math. Soc., Providence, R.I., 1969. MR 0246142 (39:7447)
  • 27. Gohberg, I., Krein, M.G., Theory and applications of Volterra operators in Hilbert space, Amer. Math. Soc., Providence, R.I., 1971. MR 0264447 (41:9041)
  • 28. Gorbachuk, M.L., Gorbachuck, V.I., M.G.Krein's Lectures on Entire Operators Birkhäuser, Basel, 1997. MR 1466698 (99f:47001)
  • 29. Hamburger, H.L., Über die Zerlegung des Hilbertschen Raumes durch vollstetige lineare Transformationen, Math. Nachr. 4 (1951), 56-69. MR 0040587 (12:718b)
  • 30. Horn, R.A., Johnson, C.R., Matrix Analysis, Cambridge Univ. Press, Cambridge, 1985. MR 0832183 (87e:15001)
  • 31. Kato, T., Perturbation theory for linear operators, Springer-Verlag, Berlin, 1995. MR 1335452 (96a:47025)
  • 32. Krein, M.G., Naimark, M.A., 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. MR 0638124 (84i:12016)
  • 33. Helgason, S., Differential geometry, Lie groups, and symmetric spaces, Academic Press, Boston, 1978. MR 0514561 (80k:53081)
  • 34. Helson, H., Large analytic functions, II., Analysis and Partial Differential Equations, Lecture Notes in Pure and Appl. Math., 122, Dekker, New York, 1990, 217-220.MR 1044789 (92c:30039)
  • 35. Lotto, B.A., Sarason, D., Multiplicative structure of de Branges's spaces, Rev. Mat. Iberoamericana 7 (1991), 183-220. MR 1133377 (92k:46035)
  • 36. Lucenko, I. E., Linear operators that commute with antiunitary operators (Russian), Teor. Funkt. Funktional. Anal. Prilozen. Vyp., 9 (1969), 85-93.MR 0278085 (43:3817)
  • 37. von Neumann, J., Allgemeine Eigenwerttheorie Hermitischer Funktionaloperatoren, Math. Ann. 102 (1929), 49-131.
  • 38. Nikolski, N.K., Treatise on the Shift Operator, Springer-Verlag, New York, 1986. MR 0827223 (87i:47042)
  • 39. Nikolski, N.K., Operators, Functions, and Systems: An Easy Reading, Volume 2: Model Operators and Systems, American Mathematical Society, Providence R.I., 2002.MR 1892647 (2003i:47001b)
  • 40. Peller, V.V., Hankel Operators and Their Applications, Springer Monographs in Mathematics, Springer-Verlag, 2003.MR 1949210 (2004e:47040)
  • 41. Potapov, V.P., The multiplicative structure of J-contractive matrix functions (in Russian), Trudy Moskov. Mat. Obsc. 4 (1955), 125-236. MR 0076882 (17:958f)
  • 42. Putinar, M., Shapiro, H.S., The Friedrichs operator of a planar domain, I., "S. A. Vinogradov Memorial Volume" (V. Havin, N. K. Nikolskii, eds.), Birkhäuser Verlag, Basel et al., 2000, 303-330; Part II., ``Béla Sz.-Nagy Memorial Volume" (L. Kérchy et al., eds.), Birkhäuser, Basel, 2001, 519-551.MR 1771771 (2001g:47049); MR 1902820 (2003e:47054)
  • 43. Reed, M., Simon, B., Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975.MR 0493419 (58:12429a)
  • 44. Richter, S., Sundberg, C., A formula for the local Dirichlet integral, Michigan Math. J. 38 (1991), 355-379.MR 1116495 (92i:47035)
  • 45. Riesz, F., Sz.-Nagy, B., Functional Analysis, Dover, New York, 1990.MR 1068530 (91g:00002)
  • 46. Ross, W.T., Shapiro, H.S., Generalized Analytic Continuation, University Lecture Series, Volume 25, American Mathematical Society, Providence R.I., 2002. MR 1895624 (2003h:30003)
  • 47. Rudin, W., Function theory in the polydisk, W.A. Benjamin, New York, 1969. MR 0255841 (41:501)
  • 48. M. Schiffer, Fredholm eigenvalues and Grunsky matrices, Ann. Polonici Math. 39 (1981), 149- 164. MR 0617457 (82k:30010)
  • 49. Siegel, C.L., Symplectic geometry, Amer. J. Math. 67 (1943), 1-86. MR 0008094 (4:242b)
  • 50. Simon, B., Spectral analysis of rank one perturbations and applications, in vol. Mathematical quantum theory. II. Schrödinger operators (Vancouver, BC, 1993), pp. 109-149, CRM Proc. Lecture Notes 8, Amer. Math. Soc., Providence, RI, 1995.MR 1332038 (97c:47008)
  • 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. Taussky, O., The role of symmetric matrices in the study of general matrices, Linear Algebra Appl. 5 (1972), 147-154.MR 0302674 (46:1818)
  • 53. Thompson, J., Approximation in the mean by polynomials, Ann. Math. 133 (1991), 477-507. MR 1109351 (93g:47026)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 30D55, 47A15

Retrieve articles in all journals with MSC (2000): 30D55, 47A15


Additional Information

Stephan Ramon Garcia
Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106-3080
Email: garcias@math.ucsb.edu

Mihai Putinar
Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106-3080
Email: mputinar@math.ucsb.edu

DOI: https://doi.org/10.1090/S0002-9947-05-03742-6
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

American Mathematical Society