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)



The backward shift on Dirichlet-type spaces

Author: Stephan Ramon Garcia
Journal: Proc. Amer. Math. Soc. 133 (2005), 3047-3056
MSC (2000): Primary 30D55, 47B38
Published electronically: March 31, 2005
MathSciNet review: 2159784
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the backward shift operator on Hilbert spaces ${\mathcal{H}}_{\alpha}$ (for ${\alpha \geq 0}$) which are norm equivalent to the Dirichlet-type spaces $D_{\alpha}$. Although these operators are unitarily equivalent to the adjoints of the forward shift operator on certain weighted Bergman spaces, our approach is direct and completely independent of the standard Cauchy duality. We employ only the classical Hardy space theory and an elementary formula expressing the inner product on ${\mathcal{H}}_{\alpha}$ in terms of a weighted superposition of backward shifts.

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

  • 1. AHERN, P., The mean modulus and derivative of an inner function, Math. J., 28, no.2 (1979), 311-347. MR 0523107 (80h:30027)
  • 2. AHLFORS, L.V., Complex Analysis (Third Edition), International Series in Pure and Applied Mathematics, McGraw-Hill, 1979. MR 0510197 (80c:30001)
  • 3. ALEMAN, A. The multiplication operator on Hilbert spaces of analytic functions, Habilitationsschrift, Hagen, 1993.
  • 4. ALEMAN, A., RICHTER, S., SUNDBERG, C., Beurling's theorem for the Bergman space, Acta Math., 177 (1996), 275-310. MR 1440934 (98a:46034)
  • 5. ALEMAN, A., RICHTER, S., ROSS, W.T., Pseudocontinuations and the backward shift, Indiana Univ. Math. J., 47 (1998), no.1, 223-276. MR 1631561 (2000i:47009)
  • 6. ALEMAN, A., RICHTER, S., Simply invariant subspaces of $H^2$ of some multiply connected regions, Integral Equations Operator Theory, 24 (1996), 127-155. MR 1371943 (99b:47010a)
  • 7. CARLESON, L., A representation theorem for the Dirichlet integral, Math. Z. 73 (1960), 190-196. MR 0112958 (22:3803)
  • 8. CIMA, J.A., ROSS, W.T., The Backward Shift on the Hardy Space, American Mathematical Society, 2000. MR 1761913 (2002f:47068)
  • 9. DOUGLAS, R.G.; SHAPIRO, H.S.; SHIELDS, A.L., Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier (Grenoble) 20, no. 1 (1970), 37-76. MR 0270196 (42:5088)
  • 10. DUREN, P., Theory of $H^p$ Spaces, Pure and Appl. Math., Vol. 38, Academic Press, 1970. MR 0268655 (42:3552)
  • 11. DUREN, P., SCHUSTER, A., Bergman Spaces, Mathematical Surveys and Monographs, Vol. 100, American Mathematical Society, 2004. MR 2033762
  • 12. GARCIA, S.R., Conjugation, the backward shift, and Toeplitz kernels, J. Operator Theory, to appear.
  • 13. HALMOS, P.R., A Hilbert Space Problem Book (Second Edition), Graduate Texts in Mathematics 19, Springer-Verlag, 1982. MR 0675952 (84e:47001)
  • 14. HEDENMALM, H., KORENBLUM, B., ZHU, K., Theory of Bergman Spaces, Graduate Texts in Mathematics 199, Springer-Verlag, 2000. MR 1758653 (2001c:46043)
  • 15. NEWMAN, D.J., SHAPIRO, H.S., The Taylor coefficients of inner functions, Michigan Math. J. 9 (1962), 249-255. MR 0148874 (26:6371)
  • 16. RICHTER, S., A representation theorem for cyclic analytic two-isometries, Trans. Amer. Math. Soc. 328, no.1 (1991), 325-349. MR 1013337 (92e:47052)
  • 17. ROSS, W.T., SHAPIRO, H.S., Generalized Analytic Continuation, University Lecture Series, Volume 25, American Mathematical Society, 2002. MR 1895624 (2003h:30003)
  • 18. SHIROKOV, N.A., Analytic Functions Smooth Up to the Boundary, Lecture Notes in Mathematics 1312, Springer-Verlag, 1988. MR 0947146 (90h:30087)
  • 19. SZ.-NAGY, B., FOIAS, C., Harmonic Analysis of Operators on Hilbert space, North-Holland, Amsterdam-London, 1970. MR 0275190 (43:947)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 30D55, 47B38

Retrieve articles in all journals with MSC (2000): 30D55, 47B38

Additional Information

Stephan Ramon Garcia
Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California, 93106-3080

Keywords: Backward shift operator, Dirichlet-type spaces, weighted Bergman spaces, cyclic function, noncyclic function, invariant subspaces, pseudocontinuation, Bergman shift operator
Received by editor(s): May 8, 2004
Received by editor(s) in revised form: May 31, 2004
Published electronically: March 31, 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.

American Mathematical Society