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An estimate of the essential norm of a composition operator from \(F(p,q,s)\) to \(\mathcal B^\alpha \) in the unit ball. (English) Zbl 1191.47033

Summary: Let \(B_n\) be the unit ball of \(\mathbb C^{n}\) and \(\varphi =(\varphi_{1},\dots ,\varphi_{n})\) a holomorphic self-map of \(B_n\). Let \(0<p,s<\infty\), \(-n-1<q<\infty\), \(q+s>-1\), \(\alpha >0\), and let \(C_{\varphi}\) be the composition operator between the space \(F(p,q,s)\) and \(\alpha \)-Bloch space \(\mathcal B^{\alpha}\) induced by \(\varphi \). This paper gives an estimate of the essential norm of \(C_{\varphi}\). As a consequence, a necessary and sufficient condition for the composition operator \(C_{\varphi}\) to be compact from \(F(p,q,s)\) to \(\mathcal B^{\alpha}\) is obtained.

MSC:

47B33 Linear composition operators
46E15 Banach spaces of continuous, differentiable or analytic functions
32A18 Bloch functions, normal functions of several complex variables
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References:

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