A $*$-closed subalgebra of the Smirnov class
HTML articles powered by AMS MathViewer
- by Stephan Ramon Garcia PDF
- Proc. Amer. Math. Soc. 133 (2005), 2051-2059 Request permission
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
- A. B. Aleksandrov, Essays on nonlocally convex Hardy classes, Complex analysis and spectral theory (Leningrad, 1979/1980) Lecture Notes in Math., vol. 864, Springer, Berlin-New York, 1981, pp. 1–89. MR 643380
- Joseph A. Cima and William T. Ross, The backward shift on the Hardy space, Mathematical Surveys and Monographs, vol. 79, American Mathematical Society, Providence, RI, 2000. MR 1761913, DOI 10.1090/surv/079
- R. G. Douglas, H. S. Shapiro, and A. L. Shields, Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier (Grenoble) 20 (1970), no. fasc. 1, 37–76 (English, with French summary). MR 270196, DOI 10.5802/aif.338
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
- Garcia, S.R., Conjugation, the backward shift, and Toeplitz kernels, (To appear: J. Operator Theory).
- Stephan Ramon Garcia and Donald Sarason, Real outer functions, Indiana Univ. Math. J. 52 (2003), no. 6, 1397–1412. MR 2021044, DOI 10.1512/iumj.2003.52.2511
- Henry Helson, Large analytic functions. II, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 217–220. MR 1044789
- Henry Helson and Donald Sarason, Past and future, Math. Scand. 21 (1967), 5–16 (1968). MR 236989, DOI 10.7146/math.scand.a-10840
- J. Neuwirth and D. J. Newman, Positive $H^{1/2}$ functions are constants, Proc. Amer. Math. Soc. 18 (1967), 958. MR 213576, DOI 10.1090/S0002-9939-1967-0213576-5
- William T. Ross and Harold S. Shapiro, Generalized analytic continuation, University Lecture Series, vol. 25, American Mathematical Society, Providence, RI, 2002. MR 1895624, DOI 10.1090/ulect/025
Additional Information
- Stephan Ramon Garcia
- Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106-3080
- MR Author ID: 726101
- Email: garcias@math.ucsb.edu
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 2051-2059
- MSC (2000): Primary 30D55
- DOI: https://doi.org/10.1090/S0002-9939-05-07735-X
- MathSciNet review: 2137871