On a Class of Composition Operators on Bergman Space
Let 𝔻={z∈ℂ:|z|<1} be the open unit disk in the complex plane ℂ. Let A2(𝔻) be the space of analytic functions on 𝔻 square integrable with respect to the measure dA(z)=(1/π)dx dy. Given a∈𝔻 and f any measurable function on 𝔻, we define the...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2007-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2007/39819 |
| Summary: | Let 𝔻={z∈ℂ:|z|<1} be the open unit
disk in the complex plane ℂ. Let A2(𝔻) be the space of analytic functions on 𝔻 square integrable
with respect to the measure dA(z)=(1/π)dx dy. Given a∈𝔻 and f any
measurable function on 𝔻, we define the function
Caf by Caf(z)=f(ϕa(z)), where ϕa∈Aut(𝔻). The map Ca is a composition operator on L2(𝔻,dA) and A2(𝔻) for all a∈𝔻. Let ℒ(A2(𝔻)) be the space of
all bounded linear operators from A2(𝔻) into itself. In this article, we have shown that CaSCa=S for all a∈𝔻 if and only if
∫𝔻S˜(ϕa(z))dA(a)=S˜(z), where S∈ℒ(A2(𝔻)) and S˜ is the Berezin symbol of S. |
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| ISSN: | 0161-1712 1687-0425 |