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 $\mathcal{E}$, the
Segal--Bargmann space
$\mathcal{H}(\mathcal{E})$ is the reproducing kernel Hilbert space
associated with the kernel $K(x,y)=\exp(\langle x,y\rangle)$ for
$x,y$ in $\mathcal{E}$. If
$\varphi:\mathcal{E}_1\rightarrow\mathcal{E}_2$ is a
mapping between two Hilbert spaces, then the composition
operator $C_{\varphi}$ is defined by $C_{\varphi}h =
h\circ\varphi$ for all $h\in\mathcal{H}(\mathcal{E}_2)$ for which
$h\circ\varphi$ belongs to $\mathcal{H}(\mathcal{E}_1)$. We
determine necessary and sufficient conditions for the boundedness and
compactness of $C_{\varphi}$. In the special case where
$\mathcal{E}_1=\mathcal{E}_2=\mathbb{C}^n$, we recover
results obtained by Carswell, MacCluer and Schuster. We also compute
the spectral radii and the essential norms of a class of
operators $C_{\varphi}$.
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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