Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Contractive multipliers from Hardy space to weighted Hardy space

Authors: Joseph A. Ball and Vladimir Bolotnikov
Journal: Proc. Amer. Math. Soc. 145 (2017), 2411-2425
MSC (2010): Primary 30E05, 47A57, 46E22
Published electronically: February 20, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown how any contractive multiplier from the Hardy space to a weighted Hardy space $ H^{2}_{\mathbf {\beta }}$ can be factored as a fixed factor composed with a classical Schur multiplier (contractive multiplier between Hardy spaces). The result is applied to get results on interpolation for a Hardy-to-weighted-Hardy contractive multiplier class, as well as a new characterization of Bergman inner functions.

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

  • [1] Joseph A. Ball and Vladimir Bolotnikov, Interpolation problems for Schur multipliers on the Drury-Arveson space: from Nevanlinna-Pick to abstract interpolation problem, Integral Equations Operator Theory 62 (2008), no. 3, 301-349. MR 2461123,
  • [2] Joseph A. Ball and Vladimir Bolotnikov, Weighted Bergman spaces: shift-invariant subspaces and input/state/output linear systems, Integral Equations Operator Theory 76 (2013), no. 3, 301-356. MR 3065298,
  • [3] Joseph A. Ball and Vladimir Bolotnikov, Weighted Hardy spaces: shift invariant and coinvariant subspaces, linear systems and operator model theory, Acta Sci. Math. (Szeged) 79 (2013), no. 3-4, 623-686. MR 3134507
  • [4] Joseph A. Ball and Vladimir Bolotnikov, On the expansive property of inner functions in weighted Hardy spaces, Complex analysis and dynamical systems VI. Part 2, Contemp. Math., vol. 667, Amer. Math. Soc., Providence, RI, 2016, pp. 47-61. MR 3511251,
  • [5] Joseph A. Ball, Israel Gohberg, and Leiba Rodman, Interpolation of rational matrix functions, Operator Theory: Advances and Applications, vol. 45, Birkhäuser Verlag, Basel, 1990. MR 1083145
  • [6] G.E. Dullerud and F. Paganini, A Course in Robust Control Theory: A Convex Approach, Texts in Applied Mathematics 36, Springer, 2000.
  • [7] Haakan Hedenmalm, Boris Korenblum, and Kehe Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000. MR 1758653
  • [8] Alexander Kheifets, The abstract interpolation problem and applications, Holomorphic spaces (Berkeley, CA, 1995) Math. Sci. Res. Inst. Publ., vol. 33, Cambridge Univ. Press, Cambridge, 1998, pp. 351-379. MR 1630655
  • [9] Anders Olofsson, An operator-valued Berezin transform and the class of $ n$-hypercontractions, Integral Equations Operator Theory 58 (2007), no. 4, 503-549. MR 2329133,
  • [10] A. Olofsson, Operator-valued Bergman inner functions as transfer functions, Algebra i Analiz 19 (2007), no. 4, 146-173; English transl., St. Petersburg Math. J. 19 (2008), no. 4, 603-623. MR 2381937,
  • [11] Marvin Rosenblum and James Rovnyak, Hardy classes and operator theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. Oxford Science Publications. MR 822228
  • [12] Walter Rudin, Principles of mathematical analysis, 3rd ed., McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. International Series in Pure and Applied Mathematics. MR 0385023
  • [13] Allen L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, Amer. Math. Soc., Providence, R.I., 1974, pp. 49-128. Math. Surveys, No. 13. MR 0361899
  • [14] Sergei Treil and Alexander Volberg, A fixed point approach to Nehari's problem and its applications, Toeplitz operators and related topics (Santa Cruz, CA, 1992) Oper. Theory Adv. Appl., vol. 71, Birkhäuser, Basel, 1994, pp. 165-186. MR 1300219

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30E05, 47A57, 46E22

Retrieve articles in all journals with MSC (2010): 30E05, 47A57, 46E22

Additional Information

Joseph A. Ball
Affiliation: Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0123

Vladimir Bolotnikov
Affiliation: Department of Mathematics, The College of William and Mary, Williamsburg, Virginia 23187-8795

Keywords: Contractive multiplier, Bergman inner function, multiplier interpolation problems
Received by editor(s): March 27, 2012
Published electronically: February 20, 2017
Communicated by: Ken Ono
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society