Spectra of composition operators on the bloch and bergman spaces |
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Authors: | Barbara MacCluer Karen Saxe |
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Institution: | (1) Department of Mathematics, Kerchof Hall, University of Virginia, 22904 Charlottesville, VA, USA;(2) Department of Mathematics, Macalester College, 1600 Grand Avenue, 55105 St. Paul, MN, USA |
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Abstract: | Whenϕ is an analytic map of the unit diskU into itself, andX is a Banach space of analytic functions onU, define the composition operatorC
ϕ
byC
ϕ
(f)=f o ϕ, forf∈X. In this paper we show how to use the Calderón theory of complex interpolation to obtain information on the spectrum ofC
ϕ
(under suitable hypotheses onϕ) acting on the Bloch spaceB and BMOA, the space of analytic functions in BMO. To do this we first obtain some results on the essential spectral radius
and spectrum ofC
ϕ
on the Bergman spacesA
pand Hardy spacesH
p,spaces which are connected toB and BMOA by the interpolation relationships A
1,B]
t
=A
pand H
1,BMOA]
t
=H
pfor 1=p(1−t). |
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Keywords: | |
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