Topological and Dynamical Structures of Composition Operators on Generalized fock Spaces
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Date
2020-08-08
Authors
Seyoum, Werkaferahu
Journal Title
Journal ISSN
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Publisher
Addis Ababa University
Abstract
In this thesis we study various properties of composition operators acting between
generalized Fock spaces Fp'
and Fq'
with weight functions ' grow faster than
the classical Gaussian weight function 1
2 jzj2 and satisfy some mild smoothness
conditions. Let be an analytic map on the complex plane. Then for p 6= q, we
have shown that the composition operator C : Fp'
! Fq'
is bounded if and only
if it is compact. This result shows a signi_cance di_erence with the analogous
result for the case when C acts between the classical Fock spaces or generalized
Fock spaces where the weight functions grow slower than the Gaussian function.
We further study some topological structure of composition operators on the
spaces. It is shown that the di_erence of two composition operators is compact if
and only if both are compact, and hence cancellation phenomenon fails to exist.
While each non-compact bounded composition operator is an isolated point, the
set of all compact composition operators forms a connected components of the
space of the operators under the operator norm topology. Moreover, Schatten
Sp(F2'
) class membership, spectra, hyponormality of the composition operators
are characterized.
We also study various dynamical structure of composition operators C on the
generalized Fock spaces Fp'
and the weighted composition operatorsW(u; ) de_ned
on the classical Fock spaces Fp. It is shown that all composition operators on
the spaces are power bounded. Several conditions characterizing uniformly mean
ergodic composition operators are provided. We have identi_ed operators W(u; )
that are power bounded and uniformly mean ergodic on the spaces, and these
properties are described in terms of easy to apply conditions which are based
merely on the values ju(0)j and ju( b
1a )j, where the numbers a and b are from
linear expansion of the symbol (z) = az + b. We have proved that composition
operators C and weighted composition operatorsW(u; ) can not be supercyclic on
their respective Fock spaces. Furthermore, the operator C ; (z) = az+b; jaj _
1 and b = 0 when jaj = 1, is cyclic if and only if an 6= a, while W(u; ) is cyclic if
and only if the corresponding composition operator is cyclic and u fails to vanish
on C. The set of periodic points of C is also determined. Conditions under which
the operators satisfy the Ritt's resolvent growth conditions are also identi_ed. In
particular, we show that a non-trivial composition operator on the Fock spaces
satis_es such growth condition if and only if it is compact.
Description
Keywords
Topological, Dynamical Structures of Composition, Operators on Generalized, Spaces