The essential norm of the integral type operators

Abstract

On the basis of the characterizations of the boundedness and compactness of the Volterra type operator $$I_{g, \varphi }$$ from mixed-norm spaces $$H(p,\, q,\, \phi )$$ to Zygmund spaces $$ \mathcal {Z}$$, the authors provide a function-theoretic estimate for the essential norm of Volterra type operator $$I_{g, \varphi }$$ by means of the definition of the essential norm of an operator and the properties of the analytic function. An estimate for the essential norm of the generalized integration operator $$\begin{aligned} I^{(n)}_{g, \varphi }f(z)=\int ^{z}_{0}f^{(n)}(\varphi (\xi ))g(\xi )d\xi , \ z\in \mathbb {D}, \end{aligned}$$from Bloch-type spaces to F(p, q, s) spaces is also obtained.