Unitary equivalence to a truncated Toeplitz operator: analytic symbols
HTML articles powered by AMS MathViewer
- by Stephan Ramon Garcia, Daniel E. Poore and William T. Ross PDF
- Proc. Amer. Math. Soc. 140 (2012), 1281-1295 Request permission
Abstract:
Unlike Toeplitz operators on $H^2$, truncated Toeplitz operators do not have a natural matricial characterization. Consequently, these operators are difficult to study numerically. In this paper we provide criteria for a matrix with distinct eigenvalues to be unitarily equivalent to a truncated Toeplitz operator having an analytic symbol. This test is constructive, and we illustrate it with several examples. As a byproduct, we also prove that every complex symmetric operator on a Hilbert space of dimension $\leq 3$ is unitarily equivalent to a direct sum of truncated Toeplitz operators.References
- Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002. MR 1882259, DOI 10.1090/gsm/044
- Sheldon Axler, Linear algebra done right, 2nd ed., Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997. MR 1482226, DOI 10.1007/b97662
- Levon Balayan and Stephan Ramon Garcia, Unitary equivalence to a complex symmetric matrix: geometric criteria, Oper. Matrices 4 (2010), no. 1, 53–76. MR 2655004, DOI 10.7153/oam-04-02
- Anton Baranov, Isabelle Chalendar, Emmanuel Fricain, Javad Mashreghi, and Dan Timotin, Bounded symbols and reproducing kernel thesis for truncated Toeplitz operators, J. Funct. Anal. 259 (2010), no. 10, 2673–2701. MR 2679022, DOI 10.1016/j.jfa.2010.05.005
- I. Chalendar, E. Fricain, and D. Timotin, On an extremal problem of Garcia and Ross, Oper. Matrices 3 (2009), no. 4, 541–546. MR 2597679, DOI 10.7153/oam-03-31
- Nicolas Chevrot, Emmanuel Fricain, and Dan Timotin, The characteristic function of a complex symmetric contraction, Proc. Amer. Math. Soc. 135 (2007), no. 9, 2877–2886. MR 2317964, DOI 10.1090/S0002-9939-07-08803-X
- Joseph A. Cima, William T. Ross, and Warren R. Wogen, Truncated Toeplitz operators on finite dimensional spaces, Oper. Matrices 2 (2008), no. 3, 357–369. MR 2440673, DOI 10.7153/oam-02-21
- Joseph A. Cima, Stephan Ramon Garcia, William T. Ross, and Warren R. Wogen, Truncated Toeplitz operators: spatial isomorphism, unitary equivalence, and similarity, Indiana Univ. Math. J. 59 (2010), no. 2, 595–620. MR 2648079, DOI 10.1512/iumj.2010.59.4097
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
- Euclid, Euclid’s Elements, Green Lion Press, Santa Fe, NM, 2002. All thirteen books complete in one volume; The Thomas L. Heath translation; Edited by Dana Densmore. MR 1932864
- Stephan Ramon Garcia, Conjugation and Clark operators, Recent advances in operator-related function theory, Contemp. Math., vol. 393, Amer. Math. Soc., Providence, RI, 2006, pp. 67–111. MR 2198373, DOI 10.1090/conm/393/07372
- S. R. Garcia, D. E. Poore, and M. K. Wyse, Unitary equivalence to a complex symmetric matrix: A modulus criterion, Oper. Matrices 5 (2011), no. 2, 273–287.
- Stephan Ramon Garcia and Mihai Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 (2006), no. 3, 1285–1315. MR 2187654, DOI 10.1090/S0002-9947-05-03742-6
- Stephan Ramon Garcia and Mihai Putinar, Complex symmetric operators and applications. II, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3913–3931. MR 2302518, DOI 10.1090/S0002-9947-07-04213-4
- Stephan Ramon Garcia and William T. Ross, A non-linear extremal problem on the Hardy space, Comput. Methods Funct. Theory 9 (2009), no. 2, 485–524. MR 2572653, DOI 10.1007/BF03321742
- S. R. Garcia and J. E. Tener, Unitary equivalence of a matrix to its transpose, J. Operator Theory (to appear). http://arxiv.org/abs/0908.2107.
- Stephan Ramon Garcia and Warren R. Wogen, Some new classes of complex symmetric operators, Trans. Amer. Math. Soc. 362 (2010), no. 11, 6065–6077. MR 2661508, DOI 10.1090/S0002-9947-2010-05068-8
- Stephan Ramon Garcia, Means of unitaries, conjugations, and the Friedrichs operator, J. Math. Anal. Appl. 335 (2007), no. 2, 941–947. MR 2345511, DOI 10.1016/j.jmaa.2007.01.094
- John B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007. MR 2261424
- R. Hartshorne, Geometry: Euclid and beyond, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2000. MR 1761093 (2001h:51001)
- Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1990. Corrected reprint of the 1985 original. MR 1084815
- Alan McIntosh, The Toeplitz-Hausdorff theorem and ellipticity conditions, Amer. Math. Monthly 85 (1978), no. 6, 475–477. MR 506368, DOI 10.2307/2320069
- 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
- Carl Pearcy, A complete set of unitary invariants for $3\times 3$ complex matrices, Trans. Amer. Math. Soc. 104 (1962), 425–429. MR 144911, DOI 10.1090/S0002-9947-1962-0144911-4
- Donald Sarason, A remark on the Volterra operator, J. Math. Anal. Appl. 12 (1965), 244–246. MR 192355, DOI 10.1016/0022-247X(65)90035-1
- Donald Sarason, Algebraic properties of truncated Toeplitz operators, Oper. Matrices 1 (2007), no. 4, 491–526. MR 2363975, DOI 10.7153/oam-01-29
- Donald Sarason, Unbounded operators commuting with restricted backward shifts, Oper. Matrices 2 (2008), no. 4, 583–601. MR 2468883, DOI 10.7153/oam-02-36
- Donald Sarason, Unbounded Toeplitz operators, Integral Equations Operator Theory 61 (2008), no. 2, 281–298. MR 2418122, DOI 10.1007/s00020-008-1588-3
- N. Sedlock, Algebras of truncated Toeplitz operators, Oper. Matrices 5 (2011), no. 2, 309–326.
- K. S. Sibirskiĭ, A minimal polynomial basis of unitary invariants of a square matrix of order three, Mat. Zametki 3 (1968), 291–295 (Russian). MR 228522
- E. Strouse, D. Timotin, and M. Zarrabi, Unitary equivalence to truncated Toeplitz operators (submitted). http://arxiv.org/abs/1011.6055
- James E. Tener, Unitary equivalence to a complex symmetric matrix: an algorithm, J. Math. Anal. Appl. 341 (2008), no. 1, 640–648. MR 2394112, DOI 10.1016/j.jmaa.2007.10.029
- Lloyd N. Trefethen and Mark Embree, Spectra and pseudospectra, Princeton University Press, Princeton, NJ, 2005. The behavior of nonnormal matrices and operators. MR 2155029
- J. Vermeer, Orthogonal similarity of a real matrix and its transpose, Linear Algebra Appl. 428 (2008), no. 1, 382–392. MR 2372597, DOI 10.1016/j.laa.2007.06.028
Additional Information
- Stephan Ramon Garcia
- Affiliation: Department of Mathematics, Pomona College, Claremont, California 91711
- MR Author ID: 726101
- Email: Stephan.Garcia@pomona.edu
- Daniel E. Poore
- Affiliation: Department of Mathematics, Pomona College, Claremont, California 91711
- Email: dep02007@mymail.pomona.edu
- William T. Ross
- Affiliation: Department of Mathematics and Computer Science, University of Richmond, Richmond, Virginia 23173
- MR Author ID: 318145
- Email: wross@richmond.edu
- Received by editor(s): December 21, 2010
- Published electronically: July 22, 2011
- Additional Notes: The first author was partially supported by National Science Foundation Grant DMS-1001614.
- Communicated by: Richard Rochberg
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 1281-1295
- MSC (2010): Primary 47A05, 47B35, 47B99
- DOI: https://doi.org/10.1090/S0002-9939-2011-11060-8
- MathSciNet review: 2869112