Description of Bloch spaces, weighted Bergman spaces and invariant subspaces, and related questions

Authors

  • Mübariz T. Garayev Dept. of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
  • Mehmet Gürdal Dept. of Mathematics, Faculty of Arts and Sciences, Süleyman Demirel University, Isparta, Turkey
  • Ulaş Yamancı Dept. of Mathematics, Faculty of Arts and Sciences, Süleyman Demirel University, Isparta, Turkey

Keywords:

Berezin symbol, Bloch space, diagonal operator, invariant subspace, weighted Bergman space.

Abstract

Let D be the unit disc of complex plane C, and H=Hol(D) the class of functions analytic in D. Recall that an f∈Hol(D) is said to belong to the Bloch space B=B(D) if
‖f‖_{B}:=sup_{z∈D}(1-|z|²)|f′(z)|<+∞.
With the norm ‖f‖=|f(0)|+‖f‖_{B}, B is Banach space. Let B₀=B₀(D) be the Bloch space which consists of all f∈B satisfying
lim_{|z|→1}(1-|z|²)|f′(z)|=0.
Here we give a new description of Bloch spaces and weighted Bergman spaces in terms of Berezin symbols of diagonal operators associated with the Taylor coefficients of their functions. We also give in terms of Berezin symbols a characterization of the multiple shift invariant subspaces of these Bloch spaces. Some other questions are also discussed.

References

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Published

08-08-2016