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Item Commutants of composition operators on the Hardy space of the disk(2013-11-06) Carter, James Michael; Cowen, Carl C.; Klimek, Slawomir; Perez, Rodrigo A.; Chin, Raymond; Bell, Steven R.; Mukhin, EvgenyThe main part of this thesis, Chapter 4, contains results on the commutant of a semigroup of operators defined on the Hardy Space of the disk where the operators have hyperbolic non-automorphic symbols. In particular, we show in Chapter 5 that the commutant of the semigroup of operators is in one-to-one correspondence with a Banach algebra of bounded analytic functions on an open half-plane. This algebra of functions is a subalgebra of the standard Newton space. Chapter 4 extends previous work done on maps with interior fixed point to the case of the symbol of the composition operator having a boundary fixed point.Item Convexity of the Berezin Range(Elsevier, 2022) Cowen, Carl C.; Felder, Christopher; Mathematical Sciences, School of ScienceThis paper discusses the convexity of the range of the Berezin transform. For a bounded operator T acting on a reproducing kernel Hilbert space H (on a set X), this is the set B(T):={〈Tkˆx,kˆx〉H:x∈X}, where kˆx is the normalized reproducing kernel for H at x∈X. Primarily, we focus on characterizing convexity of this range for a class of composition operators acting on the Hardy space of the unit disk.