Advertisement

Recent Progress on Truncated Toeplitz Operators

  • Stephan Ramon Garcia
  • William T. RossEmail author
Chapter
Part of the Fields Institute Communications book series (FIC, volume 65)

Abstract

This paper is a survey on the emerging theory of truncated Toeplitz operators. We begin with a brief introduction to the subject and then highlight the many recent developments in the field since Sarason’s seminal paper (Oper. Matrices 1(4):491–526, 2007).

Keywords

Toeplitz Operator Blaschke Product Extremal Function Toeplitz Matrix Toeplitz Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

First author partially supported by National Science Foundation Grant DMS-1001614. Second author partially supported by National Science Foundation Grant DMS-1001614.

References

  1. 1.
    Agler, J., McCarthy, J.E.: Pick Interpolation and Hilbert Function Spaces. Graduate Studies in Mathematics, vol. 44. Am. Math. Soc., Providence (2002) zbMATHGoogle Scholar
  2. 2.
    Ahern, P.R., Clark, D.N.: On functions orthogonal to invariant subspaces. Acta Math. 124, 191–204 (1970) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ahern, P.R., Clark, D.N.: Radial limits and invariant subspaces. Amer. J. Math. 92, 332–342 (1970) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Aleksandrov, A.B.: Invariant subspaces of the backward shift operator in the space H p (p∈(0, 1)). Zap. Nauchn. Sem. Leningr. Otdel. Mat. Inst. Steklov. (LOMI) 92, 7–29 (1979), also see p. 318. Investigations on linear operators and the theory of functions, IX zbMATHGoogle Scholar
  5. 5.
    Aleksandrov, A.B.: Invariant subspaces of shift operators. An axiomatic approach. Zap. Nauchn. Sem. Leningr. Otdel. Mat. Inst. Steklov. (LOMI), 113, 7–26 (1981), also see p. 264. Investigations on linear operators and the theory of functions, XI zbMATHGoogle Scholar
  6. 6.
    Aleksandrov, A.B.: On the existence of angular boundary values of pseudocontinuable functions. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 222, 5–17 (1995), also see p. 307 (Issled. po, Linein. Oper. i Teor. Funktsii. 23) Google Scholar
  7. 7.
    Aleksandrov, A.B.: Embedding theorems for coinvariant subspaces of the shift operator. II. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 262, 5–48 (1999), also see p. 231 (Issled. po, Linein. Oper. i Teor. Funkts. 27) Google Scholar
  8. 8.
    Aleman, A., Korenblum, B.: Derivation-invariant subspaces of C . Comput. Methods Funct. Theory 8(1–2), 493–512 (2008) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Aleman, A., Richter, S.: Simply invariant subspaces of H 2 of some multiply connected regions. Integral Equ. Oper. Theory 24(2), 127–155 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Arveson, W.: A Short Course on Spectral Theory. Graduate Texts in Mathematics, vol. 209. Springer, New York (2002) zbMATHGoogle Scholar
  11. 11.
    Axler, S., Conway, J.B., McDonald, G.: Toeplitz operators on Bergman spaces. Can. J. Math. 34(2), 466–483 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Balayan, L., Garcia, S.R.: Unitary equivalence to a complex symmetric matrix: geometric criteria. Oper. Matrices 4(1), 53–76 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Baranov, A., Bessonov, R., Kapustin, V.: Symbols of truncated Toeplitz operators. J. Funct. Anal. 261, 3437–3456 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Baranov, A., Chalendar, I., Fricain, E., Mashreghi, J.E., Timotin, D.: Bounded symbols and reproducing kernel thesis for truncated Toeplitz operators. J. Funct. Anal. 259(10), 2673–2701 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Basor, E.L.: Toeplitz determinants, Fisher-Hartwig symbols, and random matrices. In: Recent Perspectives in Random Matrix Theory and Number Theory. London Math. Soc. Lecture Note Ser., vol. 322, pp. 309–336. Cambridge University Press, Cambridge (2005) CrossRefGoogle Scholar
  16. 16.
    Bercovici, H.: Operator Theory and Arithmetic in H . Mathematical Surveys and Monographs, vol. 26. Am. Math. Soc., Providence (1988) zbMATHGoogle Scholar
  17. 17.
    Bercovici, H., Foias, C., Tannenbaum, A.: On skew Toeplitz operators. I. In: Topics in Operator Theory and Interpolation. Oper. Theory Adv. Appl., vol. 29, pp. 21–43. Birkhäuser, Basel (1988) CrossRefGoogle Scholar
  18. 18.
    Bercovici, H., Foias, C., Tannenbaum, A.: On skew Toeplitz operators. II. In: Nonselfadjoint Operator Algebras, Operator Theory, and Related Topics, Oper. Theory Adv. Appl., vol. 104, pp. 23–35. Birkhäuser, Basel (1998) CrossRefGoogle Scholar
  19. 19.
    Böttcher, A., Silbermann, B.: Introduction to Large Truncated Toeplitz Matrices. Universitext. Springer, New York (1999) zbMATHCrossRefGoogle Scholar
  20. 20.
    Brown, A., Halmos, P.R.: Algebraic properties of Toeplitz operators. J. Reine Angew. Math. 213, 89–102 (1963/1964) MathSciNetGoogle Scholar
  21. 21.
    Carleson, L.: Interpolations by bounded analytic functions and the corona problem. Ann. Math. (2) 76, 547–559 (1962) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Chalendar, I., Chevrot, N., Partington, J.R.: Nearly invariant subspaces for backwards shifts on vector-valued Hardy spaces. J. Oper. Theory 63(2), 403–415 (2010) MathSciNetzbMATHGoogle Scholar
  23. 23.
    Chalendar, I., Fricain, E., Timotin, D.: On an extremal problem of Garcia and Ross. Oper. Matrices 3(4), 541–546 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Chevrot, N., Fricain, E., Timotin, D.: The characteristic function of a complex symmetric contraction. Proc. Am. Math. Soc. 135(9), 2877–2886 (2007) (electronic) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Cima, J.A., Garcia, S.R., Ross, W.T., Wogen, W.R.: Truncated Toeplitz operators: spatial isomorphism, unitary equivalence, and similarity. Indiana Univ. Math. J. 59(2), 595–620 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Cima, J.A., Matheson, A.L., Ross, W.T.: The Cauchy Transform. Mathematical Surveys and Monographs, vol. 125. Am. Math. Soc., Providence (2006) zbMATHGoogle Scholar
  27. 27.
    Cima, J.A., Ross, W.T.: The Backward Shift on the Hardy Space. Mathematical Surveys and Monographs, vol. 79. Am. Math. Soc., Providence (2000) zbMATHGoogle Scholar
  28. 28.
    Cima, J.A., Ross, W.T., Wogen, W.R.: Truncated Toeplitz operators on finite dimensional spaces. Oper. Matrices 2(3), 357–369 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Clark, D.N.: One dimensional perturbations of restricted shifts. J. Anal. Math. 25, 169–191 (1972) zbMATHCrossRefGoogle Scholar
  30. 30.
    Coburn, L.A.: The C -algebra generated by an isometry. Bull. Am. Math. Soc. 73, 722–726 (1967) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Coburn, L.A.: The C -algebra generated by an isometry. II. Trans. Am. Math. Soc. 137, 211–217 (1969) MathSciNetzbMATHGoogle Scholar
  32. 32.
    Cohn, W.: Radial limits and star invariant subspaces of bounded mean oscillation. Amer. J. Math. 108(3), 719–749 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Conway, J.B.: A Course in Operator Theory. Graduate Studies in Mathematics, vol. 21. Am. Math. Soc., Providence (2000) zbMATHGoogle Scholar
  34. 34.
    Crofoot, R.B.: Multipliers between invariant subspaces of the backward shift. Pac. J. Math. 166(2), 225–246 (1994) MathSciNetzbMATHGoogle Scholar
  35. 35.
    Danciger, J., Garcia, S.R., Putinar, M.: Variational principles for symmetric bilinear forms. Math. Nachr. 281(6), 786–802 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Davidson, K.R.: C -Algebras by Example. Fields Institute Monographs, vol. 6. Am. Math. Soc., Providence (1996) Google Scholar
  37. 37.
    Davis, P.J.: Circulant Matrices. Wiley, New York (1979). A Wiley-Interscience Publication, Pure and Applied Mathematics zbMATHGoogle Scholar
  38. 38.
    Douglas, R.G.: Banach Algebra Techniques in Operator Theory, 2nd edn. Graduate Texts in Mathematics, vol. 179. Springer, New York (1998) zbMATHCrossRefGoogle Scholar
  39. 39.
    Douglas, R.G., Shapiro, H.S., Shields, A.L.: Cyclic vectors and invariant subspaces for the backward shift operator. Ann. Inst. Fourier (Grenoble) 20, 37–76 (1970) MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Duren, P.L.: Theory of H p Spaces. Academic Press, New York (1970) zbMATHGoogle Scholar
  41. 41.
    Dyakonov, K., Khavinson, D.: Smooth functions in star-invariant subspaces. In: Recent Advances in Operator-Related Function Theory. Contemp. Math., vol. 393, pp. 59–66. Am. Math. Soc., Providence (2006) CrossRefGoogle Scholar
  42. 42.
    Foias, C., Frazho, A.E.: The Commutant Lifting Approach to Interpolation Problems. Operator Theory: Advances and Applications, vol. 44. Birkhäuser, Basel (1990) zbMATHGoogle Scholar
  43. 43.
    Foias, C., Tannenbaum, A.: On the Nehari problem for a certain class of L -functions appearing in control theory. J. Funct. Anal. 74(1), 146–159 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Foias, C., Tannenbaum, A.: On the Nehari problem for a certain class of L functions appearing in control theory. II. J. Funct. Anal. 81(2), 207–218 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Garcia, S.R.: Conjugation and Clark operators. In: Recent Advances in Operator-Related Function Theory. Contemp. Math., vol. 393, pp. 67–111. Am. Math. Soc., Providence (2006) CrossRefGoogle Scholar
  46. 46.
    Garcia, S.R.: Aluthge transforms of complex symmetric operators. Integral Equ. Oper. Theory 60(3), 357–367 (2008) zbMATHCrossRefGoogle Scholar
  47. 47.
    Garcia, S.R., Poore, D.E.: On the closure of the complex symmetric operators: compact operators and weighted shifts. Preprint. arXiv:1106.4855
  48. 48.
    Garcia, S.R., Poore, D.E.: On the norm closure problem for complex symmetric operators. Proc. Am. Math. Soc., to appear. arXiv:1103.5137
  49. 49.
    Garcia, S.R., Putinar, M.: Complex symmetric operators and applications. Trans. Am. Math. Soc. 358(3), 1285–1315 (2006) (electronic) MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Garcia, S.R., Putinar, M.: Complex symmetric operators and applications. II. Trans. Am. Math. Soc. 359(8), 3913–3931 (2007) (electronic) MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Garcia, S.R., Ross, W.T.: A nonlinear extremal problem on the Hardy space. Comput. Methods Funct. Theory 9(2), 485–524 (2009) MathSciNetzbMATHGoogle Scholar
  52. 52.
    Garcia, S.R., Ross, W.T.: The norm of a truncated Toeplitz operator. CRM Proc. Lect. Notes 51, 59–64 (2010) MathSciNetGoogle Scholar
  53. 53.
    Garcia, S.R., Tener, J.E.: Unitary equivalence of a matrix to its transpose. J. Oper. Theory 68(1), 179–203 (2012) MathSciNetGoogle Scholar
  54. 54.
    Garcia, S.R., Wogen, W.R.: Complex symmetric partial isometries. J. Funct. Anal. 257(4), 1251–1260 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Garcia, S.R., Wogen, W.R.: Some new classes of complex symmetric operators. Trans. Am. Math. Soc. 362(11), 6065–6077 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Garcia, S.R., Poore, D.E., Ross, W.: Unitary equivalence to a truncated Toeplitz operator: analytic symbols. Proc. Am. Math. Soc. 140, 1281–1295 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Garcia, S.R., Poore, D.E., Tener, J.E.: Unitary equivalence to a complex symmetric matrix: low dimensions. Lin. Alg. Appl. 437, 271–284 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Garcia, S.R., Poore, D.E., Wyse, M.K.: Unitary equivalence to a complex symmetric matrix: a modulus criterion. Oper. Matrices 4(1), 53–76 (2010) MathSciNetzbMATHGoogle Scholar
  59. 59.
    Garcia, S.R., Ross, W., Wogen, W.: Spatial isomorphisms of algebras of truncated Toeplitz operators. Indiana Univ. Math. J. 59, 1971–2000 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Garcia, S.R., Ross, W., Wogen, W.: C -algebras generated by truncated Toeplitz operators. Oper. Theory. Adv. Appl., to appear Google Scholar
  61. 61.
    Garcia, S.R.: The eigenstructure of complex symmetric operators. In: Recent Advances in Matrix and Operator Theory. Oper. Theory Adv. Appl., vol. 179, pp. 169–183. Birkhäuser, Basel (2008) CrossRefGoogle Scholar
  62. 62.
    Garnett, J.: Bounded Analytic Functions, 1st edn. Graduate Texts in Mathematics, vol. 236. Springer, New York (2007) Google Scholar
  63. 63.
    Gilbreath, T.M., Wogen, W.R.: Remarks on the structure of complex symmetric operators. Integral Equ. Oper. Theory 59(4), 585–590 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Gohberg, I.C., Krupnik, N.Ja.: The algebra generated by the Toeplitz matrices. Funkc. Anal. Ego Prilož. 3(2), 46–56 (1969) MathSciNetGoogle Scholar
  65. 65.
    Hartmann, A., Ross, W.T.: Boundary values in range spaces of co-analytic truncated Toeplitz operators. Publ. Mat. 56, 191–223 (2012) MathSciNetzbMATHGoogle Scholar
  66. 66.
    Hartmann, A., Sarason, D., Seip, K.: Surjective Toeplitz operators. Acta Sci. Math. (Szeged) 70(3–4), 609–621 (2004) MathSciNetzbMATHGoogle Scholar
  67. 67.
    Heinig, G.: Not every matrix is similar to a Toeplitz matrix. In: Proceedings of the Eighth Conference of the International Linear Algebra Society, Barcelona, 1999, vol. 332/334, pp. 519–531 (2001) Google Scholar
  68. 68.
    Hitt, D.: Invariant subspaces of H 2 of an annulus. Pac. J. Math. 134(1), 101–120 (1988) MathSciNetzbMATHGoogle Scholar
  69. 69.
    Hoffman, K.: Banach Spaces of Analytic Functions. Prentice-Hall Series in Modern Analysis. Prentice-Hall, Englewood Cliffs (1962) zbMATHGoogle Scholar
  70. 70.
    Johansson, K.: Toeplitz determinants, random growth and determinantal processes. In: Proceedings of the International Congress of Mathematicians, Beijing, 2002, vol. III, pp. 53–62. Higher Ed. Press, Beijing (2002) Google Scholar
  71. 71.
    Jung, S., Ko, E., Lee, J.: On scalar extensions and spectral decompositions of complex symmetric operators. J. Math. Anal. Appl. 379, 325–333 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Jung, S., Ko, E., Lee, M., Lee, J.: On local spectral properties of complex symmetric operators. J. Math. Anal. Appl. 379, 325–333 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Kiselev, A.V., Naboko, S.N.: Nonself-adjoint operators with almost Hermitian spectrum: matrix model. I. J. Comput. Appl. Math. 194(1), 115–130 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    Li, C.G., Zhu, S., Zhou, T.: Foguel operators with complex symmetry. Preprint Google Scholar
  75. 75.
    Mackey, D.S., Mackey, N., Petrovic, S.: Is every matrix similar to a Toeplitz matrix? Linear Algebra Appl. 297(1–3), 87–105 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    Makarov, N., Poltoratski, A.: Meromorphic inner functions, Toeplitz kernels and the uncertainty principle. In: Perspectives in Analysis. Math. Phys. Stud., vol. 27, pp. 185–252. Springer, Berlin (2005) CrossRefGoogle Scholar
  77. 77.
    Nikolski, N.: Operators, Functions, and Systems: An Easy Reading. Vol. 1. Mathematical Surveys and Monographs, vol. 92 Google Scholar
  78. 78.
    Nikolski, N.: Treatise on the Shift Operator. Springer, Berlin (1986) CrossRefGoogle Scholar
  79. 79.
    Nikolski, N.: Operators, Functions, and Systems: An Easy Reading. Vol. 2 Mathematical Surveys and Monographs, vol. 93. Am. Math. Soc., Providence (2002). Model operators and systems, Translated from the French by Andreas Hartmann and revised by the author Google Scholar
  80. 80.
    Partington, J.R.: Linear Operators and Linear Systems: An Analytical Approach to Control Theory. London Mathematical Society Student Texts, vol. 60. Cambridge University Press, Cambridge (2004) zbMATHCrossRefGoogle Scholar
  81. 81.
    Peller, V.V.: Hankel Operators and Their Applications. Springer Monographs in Mathematics. Springer, New York (2003) zbMATHCrossRefGoogle Scholar
  82. 82.
    Rosenblum, M., Rovnyak, J.: Hardy Classes and Operator Theory. Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, New York (1985). Oxford Science Publications zbMATHGoogle Scholar
  83. 83.
    Ross, W.T., Shapiro, H.S.: Generalized Analytic Continuation. University Lecture Series, vol. 25. Am. Math. Soc., Providence (2002) zbMATHGoogle Scholar
  84. 84.
    Sarason, D.: A remark on the Volterra operator. J. Math. Anal. Appl. 12, 244–246 (1965) MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    Sarason, D.: Generalized interpolation in H . Trans. Am. Math. Soc. 127, 179–203 (1967) MathSciNetzbMATHGoogle Scholar
  86. 86.
    Sarason, D.: Invariant Subspaces. Topics in Operator Theory, pp. 1–47. Am. Math. Soc., Providence (1974). Math. Surveys, No. 13 Google Scholar
  87. 87.
    Sarason, D.: Nearly invariant subspaces of the backward shift. In: Contributions to Operator Theory and Its Applications, Mesa, AZ, 1987. Oper. Theory Adv. Appl., vol. 35, pp. 481–493. Birkhäuser, Basel (1988) CrossRefGoogle Scholar
  88. 88.
    Sarason, D.: Algebraic properties of truncated Toeplitz operators. Oper. Matrices 1(4), 491–526 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  89. 89.
    Sarason, D.: Unbounded operators commuting with restricted backward shifts. Oper. Matrices 2(4), 583–601 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    Sarason, D.: Unbounded Toeplitz operators. Integral Equ. Oper. Theory 61(2), 281–298 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  91. 91.
    Sarason, D.: Commutant lifting. In: A Glimpse at Hilbert Space Operators. Oper. Theory Adv. Appl., vol. 207, pp. 351–357. Birkhäuser, Basel (2010) CrossRefGoogle Scholar
  92. 92.
    Sedlock, N.: Properties of truncated Toeplitz operators. Ph.D. Thesis, Washington University in St. Louis, ProQuest LLC, Ann Arbor, MI (2010) Google Scholar
  93. 93.
    Sedlock, N.: Algebras of truncated Toeplitz operators. Oper. Matrices 5(2), 309–326 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  94. 94.
    Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 1 American Mathematical Society Colloquium Publications, vol. 54. Am. Math. Soc., Providence (2005). Classical theory. MR 2105088 (2006a:42002a) zbMATHGoogle Scholar
  95. 95.
    Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 2 American Mathematical Society Colloquium Publications, vol. 54. Am. Math. Soc., Providence (2005). Spectral theory. MR 2105089 (2006a:42002b) zbMATHGoogle Scholar
  96. 96.
    Strouse, E., Timotin, D., Zarrabi, M.: Unitary equivalence to truncated Toeplitz operators. Indiana U. Math. J., to appear. http://arxiv.org/abs/1011.6055
  97. 97.
    Sz.-Nagy, B., Foias, C., Bercovici, H., Kérchy, L.: Harmonic Analysis of Operators on Hilbert Space, 2nd edn. Universitext. Springer, New York (2010) zbMATHCrossRefGoogle Scholar
  98. 98.
    Takenaka, S.: On the orthonormal functions and a new formula of interpolation. Jpn. J. Math. 2, 129–145 (1925) Google Scholar
  99. 99.
    Tener, J.E.: Unitary equivalence to a complex symmetric matrix: an algorithm. J. Math. Anal. Appl. 341(1), 640–648 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  100. 100.
    Trefethen, L.N., Embree, M.: Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton (2005) zbMATHGoogle Scholar
  101. 101.
    Vermeer, J.: Orthogonal similarity of a real matrix and its transpose. Linear Algebra Appl. 428(1), 382–392 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  102. 102.
    Volberg, A.L., Treil, S.R.: Embedding theorems for invariant subspaces of the inverse shift operator. Zap. Nauchn. Sem. Leningr. Otdel. Mat. Inst. Steklov. (LOMI) 149, 38–51 (1986), also see pp. 186–187 (Issled. Linein. Teor. Funktsii. XV) Google Scholar
  103. 103.
    Wang, X., Gao, Z.: A note on Aluthge transforms of complex symmetric operators and applications. Integral Equ. Oper. Theory 65(4), 573–580 (2009) zbMATHCrossRefGoogle Scholar
  104. 104.
    Wang, X., Gao, Z.: Some equivalence properties of complex symmetric operators. Math. Pract. Theory 40(8), 233–236 (2010) MathSciNetzbMATHGoogle Scholar
  105. 105.
    Zagorodnyuk, S.M.: On a J-polar decomposition of a bounded operator and matrix representations of J-symmetric, J-skew-symmetric operators. Banach J. Math. Anal. 4(2), 11–36 (2010) MathSciNetzbMATHGoogle Scholar
  106. 106.
    Zhu, S., Li, C.G.: Complex symmetric weighted shifts. Trans. Am. Math. Soc., to appear Google Scholar
  107. 107.
    Zhu, S., Li, C., Ji, Y.: The class of complex symmetric operators is not norm closed. Proc. Am. Math. Soc. 140, 1705–1708 (2012) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsPomona CollegeClaremontUSA
  2. 2.Department of Mathematics and Computer ScienceUniversity of RichmondRichmondUSA

Personalised recommendations