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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 $C_{\varphi}$.

Keywords:  Segal-Bargmann spaces, composition operators, positive semidefinite kernels, reproducing kernel Hilbert spaces

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