Proceedings of the Japan Academy, Series A, Mathematical Sciences

Invariant subspaces of certain sub Hilbert spaces of $H^{2}$

Abstract

Recently Yousefi and Hesameddini [13] have obtained a characterization for shift invariant subspaces of a special class of Hilbert spaces contained in the Hardy space $H^2$. In the present note we settle an open problem posed by them in their paper. In fact, by discussing invariance under multiplication by finite Blaschke factors we prove a far more general result than the main result of [13]. We prove our results under much weaker assumptions than the assumptions of [13] and with a simpler proof.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 87, Number 4 (2011), 56-59.

Dates
First available in Project Euclid: 26 April 2011

https://projecteuclid.org/euclid.pja/1303823880

Digital Object Identifier
doi:10.3792/pjaa.87.56

Mathematical Reviews number (MathSciNet)
MR2803900

Zentralblatt MATH identifier
1233.47027

Citation

Sahni, Niteesh; Singh, Dinesh. Invariant subspaces of certain sub Hilbert spaces of $H^{2}$. Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 4, 56--59. doi:10.3792/pjaa.87.56. https://projecteuclid.org/euclid.pja/1303823880

References

• A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1948), 239–255.
• P. L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38 Academic Press, New York, 1970.
• J. B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, 96, Academic Press, New York, 1981.
• K. Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis Prentice Hall, Englewood Cliffs, NJ, 1962.
• P. Koosis, Introduction to $H_{p}$ spaces, London Mathematical Society Lecture Note Series, 40, Cambridge Univ. Press, Cambridge, 1980.
• S. Lata, M. Mittal and D. Singh, A finite multiplicity Helson-Lowdenslager-de Branges theorem, Studia Math. 200 (2010), no. 3, 247–266.
• V. I. Paulsen and D. Singh, A Helson-Lowdenslager-de Branges theorem in $L^{2}$, Proc. Amer. Math. Soc. 129 (2001), no. 4, 1097–1103.
• D. A. Redett, Sub-Lebesgue Hilbert spaces on the unit circle, Bull. London Math. Soc. 37 (2005), no. 5, 793–800.
• D. Sarason, Shift-invariant spaces from the Brangesian point of view, in The Bieberbach conjecture (West Lafayette, Ind., 1985), 153–166, Math. Surveys Monogr., 21 Amer. Math. Soc., Providence, RI.
• H. S. Shapiro, Reproducing kernels and Beurling's theorem, Trans. Amer. Math. Soc. 110 (1964), 448–458.
• D. Singh and U. N. Singh, On a theorem of de Branges, Indian J. Math. 33 (1991), no. 1, 1–5.
• D. Singh and V. Thukral, Multiplication by finite Blaschke factors on de Branges spaces, J. Operator Theory 37 (1997), no. 2, 223–245.
• B. Yousefi and E. Hesameddini, Extension of the Beurling's Theorem, Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 9, 167–169.