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Recent Progress on Truncated Toeplitz Operators

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Blaschke Products and Their Applications

Part of the book series: Fields Institute Communications ((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).

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Notes

  1. 1.

    Equivalently, \(G(1/z) \in H^{p}_{0}\).

  2. 2.

    Written as an integral transform, P u can be regarded as an operator from L 1 into \(\mbox{Hol}(\mathbb{D})\).

  3. 3.

    Recall that we are using the notation \(\widetilde{f} := C f\) for \(f \in\mathcal {K}_{u}\).

References

  1. Agler, J., McCarthy, J.E.: Pick Interpolation and Hilbert Function Spaces. Graduate Studies in Mathematics, vol. 44. Am. Math. Soc., Providence (2002)

    MATH  Google Scholar 

  2. Ahern, P.R., Clark, D.N.: On functions orthogonal to invariant subspaces. Acta Math. 124, 191–204 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ahern, P.R., Clark, D.N.: Radial limits and invariant subspaces. Amer. J. Math. 92, 332–342 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  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

    MATH  Google Scholar 

  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

    MATH  Google Scholar 

  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. 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. Aleman, A., Korenblum, B.: Derivation-invariant subspaces of C ∞. Comput. Methods Funct. Theory 8(1–2), 493–512 (2008)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Arveson, W.: A Short Course on Spectral Theory. Graduate Texts in Mathematics, vol. 209. Springer, New York (2002)

    MATH  Google Scholar 

  11. Axler, S., Conway, J.B., McDonald, G.: Toeplitz operators on Bergman spaces. Can. J. Math. 34(2), 466–483 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  12. Balayan, L., Garcia, S.R.: Unitary equivalence to a complex symmetric matrix: geometric criteria. Oper. Matrices 4(1), 53–76 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Baranov, A., Bessonov, R., Kapustin, V.: Symbols of truncated Toeplitz operators. J. Funct. Anal. 261, 3437–3456 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  16. Bercovici, H.: Operator Theory and Arithmetic in H ∞. Mathematical Surveys and Monographs, vol. 26. Am. Math. Soc., Providence (1988)

    MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  19. Böttcher, A., Silbermann, B.: Introduction to Large Truncated Toeplitz Matrices. Universitext. Springer, New York (1999)

    Book  MATH  Google Scholar 

  20. Brown, A., Halmos, P.R.: Algebraic properties of Toeplitz operators. J. Reine Angew. Math. 213, 89–102 (1963/1964)

    MathSciNet  Google Scholar 

  21. Carleson, L.: Interpolations by bounded analytic functions and the corona problem. Ann. Math. (2) 76, 547–559 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  23. Chalendar, I., Fricain, E., Timotin, D.: On an extremal problem of Garcia and Ross. Oper. Matrices 3(4), 541–546 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Cima, J.A., Matheson, A.L., Ross, W.T.: The Cauchy Transform. Mathematical Surveys and Monographs, vol. 125. Am. Math. Soc., Providence (2006)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  28. Cima, J.A., Ross, W.T., Wogen, W.R.: Truncated Toeplitz operators on finite dimensional spaces. Oper. Matrices 2(3), 357–369 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Clark, D.N.: One dimensional perturbations of restricted shifts. J. Anal. Math. 25, 169–191 (1972)

    Article  MATH  Google Scholar 

  30. Coburn, L.A.: The C ∗-algebra generated by an isometry. Bull. Am. Math. Soc. 73, 722–726 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  31. Coburn, L.A.: The C ∗-algebra generated by an isometry. II. Trans. Am. Math. Soc. 137, 211–217 (1969)

    MathSciNet  MATH  Google Scholar 

  32. Cohn, W.: Radial limits and star invariant subspaces of bounded mean oscillation. Amer. J. Math. 108(3), 719–749 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  33. Conway, J.B.: A Course in Operator Theory. Graduate Studies in Mathematics, vol. 21. Am. Math. Soc., Providence (2000)

    MATH  Google Scholar 

  34. Crofoot, R.B.: Multipliers between invariant subspaces of the backward shift. Pac. J. Math. 166(2), 225–246 (1994)

    MathSciNet  MATH  Google Scholar 

  35. Danciger, J., Garcia, S.R., Putinar, M.: Variational principles for symmetric bilinear forms. Math. Nachr. 281(6), 786–802 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Davidson, K.R.: C ∗-Algebras by Example. Fields Institute Monographs, vol. 6. Am. Math. Soc., Providence (1996)

    Google Scholar 

  37. Davis, P.J.: Circulant Matrices. Wiley, New York (1979). A Wiley-Interscience Publication, Pure and Applied Mathematics

    MATH  Google Scholar 

  38. Douglas, R.G.: Banach Algebra Techniques in Operator Theory, 2nd edn. Graduate Texts in Mathematics, vol. 179. Springer, New York (1998)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  40. Duren, P.L.: Theory of H p Spaces. Academic Press, New York (1970)

    MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  42. Foias, C., Frazho, A.E.: The Commutant Lifting Approach to Interpolation Problems. Operator Theory: Advances and Applications, vol. 44. Birkhäuser, Basel (1990)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  46. Garcia, S.R.: Aluthge transforms of complex symmetric operators. Integral Equ. Oper. Theory 60(3), 357–367 (2008)

    Article  MATH  Google Scholar 

  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. 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. Garcia, S.R., Putinar, M.: Complex symmetric operators and applications. Trans. Am. Math. Soc. 358(3), 1285–1315 (2006) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  50. Garcia, S.R., Putinar, M.: Complex symmetric operators and applications. II. Trans. Am. Math. Soc. 359(8), 3913–3931 (2007) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  51. Garcia, S.R., Ross, W.T.: A nonlinear extremal problem on the Hardy space. Comput. Methods Funct. Theory 9(2), 485–524 (2009)

    MathSciNet  MATH  Google Scholar 

  52. Garcia, S.R., Ross, W.T.: The norm of a truncated Toeplitz operator. CRM Proc. Lect. Notes 51, 59–64 (2010)

    MathSciNet  Google Scholar 

  53. Garcia, S.R., Tener, J.E.: Unitary equivalence of a matrix to its transpose. J. Oper. Theory 68(1), 179–203 (2012)

    MathSciNet  Google Scholar 

  54. Garcia, S.R., Wogen, W.R.: Complex symmetric partial isometries. J. Funct. Anal. 257(4), 1251–1260 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  55. Garcia, S.R., Wogen, W.R.: Some new classes of complex symmetric operators. Trans. Am. Math. Soc. 362(11), 6065–6077 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  59. Garcia, S.R., Ross, W., Wogen, W.: Spatial isomorphisms of algebras of truncated Toeplitz operators. Indiana Univ. Math. J. 59, 1971–2000 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  60. Garcia, S.R., Ross, W., Wogen, W.: C ∗-algebras generated by truncated Toeplitz operators. Oper. Theory. Adv. Appl., to appear

    Google Scholar 

  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)

    Chapter  Google Scholar 

  62. Garnett, J.: Bounded Analytic Functions, 1st edn. Graduate Texts in Mathematics, vol. 236. Springer, New York (2007)

    Google Scholar 

  63. Gilbreath, T.M., Wogen, W.R.: Remarks on the structure of complex symmetric operators. Integral Equ. Oper. Theory 59(4), 585–590 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  64. Gohberg, I.C., Krupnik, N.Ja.: The algebra generated by the Toeplitz matrices. Funkc. Anal. Ego Prilož. 3(2), 46–56 (1969)

    MathSciNet  Google Scholar 

  65. Hartmann, A., Ross, W.T.: Boundary values in range spaces of co-analytic truncated Toeplitz operators. Publ. Mat. 56, 191–223 (2012)

    MathSciNet  MATH  Google Scholar 

  66. Hartmann, A., Sarason, D., Seip, K.: Surjective Toeplitz operators. Acta Sci. Math. (Szeged) 70(3–4), 609–621 (2004)

    MathSciNet  MATH  Google Scholar 

  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. Hitt, D.: Invariant subspaces of H 2 of an annulus. Pac. J. Math. 134(1), 101–120 (1988)

    MathSciNet  MATH  Google Scholar 

  69. Hoffman, K.: Banach Spaces of Analytic Functions. Prentice-Hall Series in Modern Analysis. Prentice-Hall, Englewood Cliffs (1962)

    MATH  Google Scholar 

  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. Jung, S., Ko, E., Lee, J.: On scalar extensions and spectral decompositions of complex symmetric operators. J. Math. Anal. Appl. 379, 325–333 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  74. Li, C.G., Zhu, S., Zhou, T.: Foguel operators with complex symmetry. Preprint

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  77. Nikolski, N.: Operators, Functions, and Systems: An Easy Reading. Vol. 1. Mathematical Surveys and Monographs, vol. 92

    Google Scholar 

  78. Nikolski, N.: Treatise on the Shift Operator. Springer, Berlin (1986)

    Book  Google Scholar 

  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. 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)

    Book  MATH  Google Scholar 

  81. Peller, V.V.: Hankel Operators and Their Applications. Springer Monographs in Mathematics. Springer, New York (2003)

    Book  MATH  Google Scholar 

  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

    MATH  Google Scholar 

  83. Ross, W.T., Shapiro, H.S.: Generalized Analytic Continuation. University Lecture Series, vol. 25. Am. Math. Soc., Providence (2002)

    MATH  Google Scholar 

  84. Sarason, D.: A remark on the Volterra operator. J. Math. Anal. Appl. 12, 244–246 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  85. Sarason, D.: Generalized interpolation in H ∞. Trans. Am. Math. Soc. 127, 179–203 (1967)

    MathSciNet  MATH  Google Scholar 

  86. Sarason, D.: Invariant Subspaces. Topics in Operator Theory, pp. 1–47. Am. Math. Soc., Providence (1974). Math. Surveys, No. 13

    Google Scholar 

  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)

    Chapter  Google Scholar 

  88. Sarason, D.: Algebraic properties of truncated Toeplitz operators. Oper. Matrices 1(4), 491–526 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  89. Sarason, D.: Unbounded operators commuting with restricted backward shifts. Oper. Matrices 2(4), 583–601 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  90. Sarason, D.: Unbounded Toeplitz operators. Integral Equ. Oper. Theory 61(2), 281–298 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  91. Sarason, D.: Commutant lifting. In: A Glimpse at Hilbert Space Operators. Oper. Theory Adv. Appl., vol. 207, pp. 351–357. Birkhäuser, Basel (2010)

    Chapter  Google Scholar 

  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. Sedlock, N.: Algebras of truncated Toeplitz operators. Oper. Matrices 5(2), 309–326 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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. Sz.-Nagy, B., Foias, C., Bercovici, H., Kérchy, L.: Harmonic Analysis of Operators on Hilbert Space, 2nd edn. Universitext. Springer, New York (2010)

    Book  MATH  Google Scholar 

  98. Takenaka, S.: On the orthonormal functions and a new formula of interpolation. Jpn. J. Math. 2, 129–145 (1925)

    Google Scholar 

  99. Tener, J.E.: Unitary equivalence to a complex symmetric matrix: an algorithm. J. Math. Anal. Appl. 341(1), 640–648 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  100. Trefethen, L.N., Embree, M.: Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton (2005)

    MATH  Google Scholar 

  101. Vermeer, J.: Orthogonal similarity of a real matrix and its transpose. Linear Algebra Appl. 428(1), 382–392 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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. Wang, X., Gao, Z.: A note on Aluthge transforms of complex symmetric operators and applications. Integral Equ. Oper. Theory 65(4), 573–580 (2009)

    Article  MATH  Google Scholar 

  104. Wang, X., Gao, Z.: Some equivalence properties of complex symmetric operators. Math. Pract. Theory 40(8), 233–236 (2010)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  106. Zhu, S., Li, C.G.: Complex symmetric weighted shifts. Trans. Am. Math. Soc., to appear

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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

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  1. Department of Mathematics, Pomona College, Claremont, CA, 91711, USA

    Stephan Ramon Garcia

  2. Department of Mathematics and Computer Science, University of Richmond, Richmond, VA, 23173, USA

    William T. Ross

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  1. , Département de mathématiques et de stati, Université Laval, avenue de la Médecine 1045, Quebec, G1K 7P4, Québec, Canada

    Javad Mashreghi

  2. , Institut Camille Jordan, Université Claude Bernard Lyon 1, avenue du 11 novembre 1918 43, Villeurbanne cedex, 69622, France

    Emmanuel Fricain

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Garcia, S.R., Ross, W.T. (2013). Recent Progress on Truncated Toeplitz Operators. In: Mashreghi, J., Fricain, E. (eds) Blaschke Products and Their Applications. Fields Institute Communications, vol 65. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-5341-3_15

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