ZERO-BASED INVARIANT SUBSPACES IN THE BERGMAN SPACE

Year 2018, Volume: 67 Issue: 1, 277 - 285, 01.02.2018

Bouabdallah Fatıha Bendaoud Zohra

https://doi.org/10.1501/Commua1_0000000849

Cited By: 1

Abstract

It is known that Beurling’s theorem concerning invariant subspaces is not true in the Bergman space (in contrast to the Hardy space case).However, Aleman, Richter, and Sundberge proved that every cyclic invariantasubspace in the Bergman space Lp(D), 0 < p < +1, is generated by its extremal function. This implies, in particular, that for every zero-based invariantsubspace in the Bergman space the Beurling’s theorem stands true. Here, wecalculate the reproducing kernel of the zero-based invariant subspace Mninathe Bergman space L2(D) where the associated wandering subspace Mnis one-dimensional, and spanned by the unit vector Gn(z) =zMn p n + 1zn

Keywords

Bergman space, inner function, Beurling’s theorem, kernel function

References

• Abkar, A., A Beuling-type theorem in Bergman spaces, Turk J Math (2011), 35, 711-716.
• Apostol, C., Bercovici, H., Foias, C. and Pearcy, C. , Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra, I
• J.Funct. Anal, (1985), 63:3, 369-404.
• Aleman, A., Richter, S. and Sundberg, C., Beurling’s theorem for the Bergman space. Acta Math (1996), 177, 275-310.
• Duren, P., Khavinson, D., Shapiro, H. S. andSundberg, C., Contractive zero-divisors in Bergman spaces, Paci…c J. Math, (1993), 157, 37-56.
• Hedenmalm, H., Resent progress in the function theory of the Bergman space. Holomorphic spaces MSRI publications (1998), 33, 35-50.
• Hedenmalm, H., A factoring theorem for the Bergman space, Bull. London Math. Soc, (1994), 26, 113-126.
• Hedenmalm, H., Korenblum, B. and Zhu, K., Theory of Bergman spaces, New York, Springerâe“Verlag, Graduate Texts in Mathematics (2000), 199.
• Khavinson, D. and Shapiro, H. S., Invariant subspaces in Bergman spaces and Hedenmalm’s boundary value problem, Ark. Mat, (1994), 32, 309-321 .
• MacLane, G. R., Holomorphic functions, of arbitrarily slow growth, without radial limits, Michigan Math, (1962), 9, 21-24. Current address : Bouabdallah Fatiha: Laboratory of Pure and Applied Mathematics, Laghouat
• University, ALGERIA E-mail address : [e-mail] ORCID: http://orcid.org/0000-0002-4890-3386 Current address : Bendaoud Zohra: Laboratory of Pure and Applied Mathematics, Laghouat
• University, ALGERIA E-mail address : [e-mail]