New York Journal of Mathematics
Volume 21 (2015) 339-350

  

Hocine Guediri, Mubariz T. Garayev, and Houcine Sadraoui

The Bergman space as a Banach algebra

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Published: May 27, 2015
Keywords: Duhamel product, Bergman space, extended eigenvalues, extended eigenoperators, cyclic vectors, intertwining relations, Banach ∗-algebra.
Subject: Primary 47B47; Secondary 47B38.

Abstract
In this paper we use the Duhamel product to provide a Banach algebra structure to each of a scale of Bergman spaces over the unit disk, and then carry out many interesting consequences. In particular we characterize cyclic vectors of the Volterra integration operator, and determine its extended eigenvalues and corresponding extended eigenoperators. We also identify its commutants and point out some intertwining relations between the Volterra integration operator and composition operators.

Acknowledgements

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project no. RGP-VPP-323.


Author information

Hocine Guediri:
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia.
hguediri@ksu.edu.sa

Mubariz T. Garayev:
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia.
mgarayev@ksu.edu.sa

Houcine Sadraoui:
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia.
sadrawi@ksu.edu.sa