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Invariant subspaces for a class of complete Pick kernels


Author: Michael T. Jury
Journal: Proc. Amer. Math. Soc. 133 (2005), 3589-3596
MSC (2000): Primary 47B32; Secondary 47A15, 47A16
DOI: https://doi.org/10.1090/S0002-9939-05-07940-2
Published electronically: June 28, 2005
MathSciNet review: 2163594
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Abstract: Motivated by the work of McCullough and Trent, we investigate the $z$-invariant subspaces of the Hilbert function spaces associated to the Szego kernels on the open unit disk. In particular, we characterize those kernels for which the the $z$-invariant subspaces are hyperinvariant, and (partially) those for which the so-called BLH subspaces are cyclic, obtaining counterexamples to two questions posed by McCullough and Trent.


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

  • 1. Agler, J. and McCarthy, J. E.. Pick interpolation and Hilbert function spaces, American Mathematical Society, Providence, RI, 2002. MR 1882259 (2003b:47001)
  • 2. Agler, J. and McCarthy, J. E., Complete Nevanlinna-Pick kernels, Journal of Functional Analysis 175 (2000), 111-124. MR 1774853 (2001h:47019)
  • 3. Aleman, A., Richter, S., and Sundberg, C., The majorization function and the index of invariant subspaces of the Bergman space, Journal Analyse de Mathématique 86 (2002), 139-182. MR 1894480 (2003g:30058)
  • 4. Arveson, W., The curvature invariant of a Hilbert module over $\mathbb{C} [z_1, \cdots, z_d]$, J. Reine. Angew. Math. 522 (2000), 173-236. MR 1758582 (2003a:47013)
  • 5. Ball, J., Trent, T., and Vinnikov, V., Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces, in Operator Theory and Analysis (Amsterdam 1997), OT122, 89-138, Birkhäuser, 2001. MR 1846055 (2002f:47028)
  • 6. Cowen, M. J. and Douglas, R. G., Complex geometry and operator theory, Acta Mathematica 141 (1978), 187-261. MR 0501368 (80f:47012)
  • 7. Greene, D., Richter, S., and Sundberg, C., The structure of inner multipliers on spaces with complete Nevanlinna-Pick kernels, Journal of Functional Analysis 194 (2002), 311-331. MR 1934606 (2003h:46038)
  • 8. Hoffman, K., Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, NJ, 1962. MR 0133008 (24:A2844)
  • 9. McCullough, S., The local deBranges-Rovnyak construction and complete Nevanlinna-Pick kernels, in Algebraic Methods in Operator Theory, 15-24, Birkhäuser, 1994. MR 1284929 (95j:47016)
  • 10. McCullough, S., and Trent, T., Invariant subspaces and Nevanlinna-Pick kernels, Journal of Functional Analysis 178 (2000), 226-249. MR 1800795 (2002b:47006)
  • 11. Quiggin, P., For which reproducing kernel Hilbert spaces is Pick's theorem true?, Integral Equations and Operator Theory 16 (1993), 244-266. MR 1205001 (94a:47026)
  • 12. -, Generalisations of Pick's theorem to reproducing kernel Hilbert spaces, Ph.D. thesis, Lancaster University, 1994.
  • 13. Sarason, D., Invariant subspaces and unstarred operator algebras, Pacific Journal of Mathematics 17 (1966), 511-517. MR 0192365 (33:590)
  • 14. -, Weak-star generators of $H^{\infty}$, Pacific Journal of Mathematics 17 (1966), 519-528. MR 0211269 (35:2151)
  • 15. -, Generalized interpolation in $H^{\infty}$, American Mathematical Society Transactions 27 (1967), 180-203. MR 0208383 (34:8193)
  • 16. Solomyak, B. M. and Volberg, A. L., Multiplicity of analytic Toeplitz operators, in Toeplitz operators and spectral function theory, 87-192, Birkhäuser, 1989. MR 1030051 (91g:47021)
  • 17. Thomson, J. E., The commutant of a class of analytic Toeplitz operators II, Indiana University Mathematics Journal 25 (1976), 793-800. MR 0417843 (54:5891)

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Additional Information

Michael T. Jury
Affiliation: Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, Indiana 47907-2067
Email: jury@math.purdue.edu

DOI: https://doi.org/10.1090/S0002-9939-05-07940-2
Keywords: Pick kernel, invariant space, cyclic vector
Received by editor(s): July 14, 2000
Received by editor(s) in revised form: July 16, 2004
Published electronically: June 28, 2005
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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