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)

 
 

 

A $*$-closed subalgebra of the Smirnov class


Author: Stephan Ramon Garcia
Journal: Proc. Amer. Math. Soc. 133 (2005), 2051-2059
MSC (2000): Primary 30D55
DOI: https://doi.org/10.1090/S0002-9939-05-07735-X
Published electronically: January 14, 2005
MathSciNet review: 2137871
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study real Smirnov functions and investigate a certain $*$-closed subalgebra of the Smirnov class $N^+$ containing them. Motivated by a result of Aleksandrov, we provide an explicit representation for the space $H^p \cap \overline{H^p}$. This leads to a natural analog of the Riesz projection on a certain quotient space of $L^p$ for $p \in (0,1)$. We also study a Herglotz-like integral transform for singular measures on the unit circle $\partial \mathbb{D} $.


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

  • 1. ALEKSANDROV, A.B, Essays on non-locally convex Hardy spaces, ``Complex Analysis and Spectral Theory," Lecture Notes in Mathematics 864, V. P. Havin and N. K. Nikol'skii (editors), Springer-Verlag, 1980. MR 0643380 (84h:46066)
  • 2. CIMA, J.A., ROSS, W.T., The Backward Shift on the Hardy Space, American Mathematical Society, 2000.MR 1761913 (2002f:47068)
  • 3. 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)
  • 4. DUREN, P.L., Theory of $H^p$ Spaces, Pure and Appl. Math., Vol 38, Academic Press, New York, 1970.MR 0268655 (42:3552)
  • 5. GARCIA, S.R., Conjugation, the backward shift, and Toeplitz kernels, (To appear: J. Operator Theory).
  • 6. GARCIA, S.R., SARASON, D., Real outer functions, Indiana Univ. Math. J. 52 (2003), 1397-1412. MR 2021044
  • 7. HELSON, H., Large analytic functions II. Analysis and Partial Differential Equations, C. Sadosky (ed.), Marcel Dekker, New York (1990), pp. 217-220. MR 1044789 (92c:30039)
  • 8. HELSON, H., SARASON, D., Past and future, Math. Scand. 21 (1967), 5-16.MR 0236989 (38:5282)
  • 9. NEUWIRTH, J., NEWMAN, D.J., Positive $H^{\frac{1}{2}}$ functions are constants, Proc. Amer. Math. Soc. 18 (1967), 968.MR 0213576 (35:4436)
  • 10. ROSS, W.T., SHAPIRO, H.S., Generalized Analytic Continuation, University Lecture Series, Volume 25, American Mathematical Society, Providence, 2002.MR 1895624 (2003h:30003)

Similar Articles

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

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


Additional Information

Stephan Ramon Garcia
Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106-3080
Email: garcias@math.ucsb.edu

DOI: https://doi.org/10.1090/S0002-9939-05-07735-X
Keywords: Riesz projection, Smirnov class, Herglotz integral, singular measure, Beurling's theorem, backward shift, pseudocontinuation
Received by editor(s): February 3, 2004
Received by editor(s) in revised form: March 5, 2004
Published electronically: January 14, 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