Journal of Operator Theory
Volume 78, Issue 1, Summer 2017 pp. 135-158.
Composition operators between Segal-Bargmann
spaces
Authors:
Trieu Le
Summary: For an arbitrary Hilbert space E, the
Segal--Bargmann space
H(E) is the reproducing kernel Hilbert space
associated with the kernel K(x,y)=exp(⟨x,y⟩) for
x,y in E. If
φ:E1→E2 is a
mapping between two Hilbert spaces, then the composition
operator Cφ is defined by Cφh=h∘φ for all h∈H(E2) for which
h∘φ belongs to H(E1). We
determine necessary and sufficient conditions for the boundedness and
compactness of Cφ. In the special case where
E1=E2=Cn, we recover
results obtained by Carswell, MacCluer and Schuster. We also compute
the spectral radii and the essential norms of a class of
operators Cφ.
DOI: http://dx.doi.org/10.7900/jot.2016jun10.2102
Keywords: Segal-Bargmann spaces, composition operators, positive
semidefinite kernels, reproducing kernel Hilbert spaces
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