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

#### 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.

‖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.

#### Keywords

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

#### Full Text:

PDF#### References

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