We study the generalized Volterra-type integral and composition operators acting on the classical Fock spaces. We first characterize various properties of the operators in terms of growth and integrability conditions which are simpler to apply than those already known Berezin type characterizations. Then, we apply these conditions to study the compact and Schatten S p class difference topological structures of the space of the operators. In particular, we proved that the difference of two Volterra-type integral operators is compact if and only if both are compact. introductionEver since A. Aleman and A. Siskakis published their seminal works [3, 4] on integral operators acting on Hardy and Bergman spaces, Volterra-type integral operators have been studied on many Banach spaces over several domains. For holomorphic functions f and g in a given domain, we define the Volterra-type integral operator V g and its companion J g byApplying integration by parts in any one of the above integrals gives the relationwhere M g f = gf is the multiplication operator of symbol g. Studies on these operators have been mainly aiming to descsribe the connection between their operator-theoretic behaviours with the function-theoretic properties of the inducing symbols g. For more information on the subject, we refer to [1,2,11] and the related references therein. In 2008, S. Li and S. Stević extended V g and J g to the operatorsand studied some of their operator-theoretic properties in terms of properties of the pairs (g, ψ) on some spaces of analytic functions on the unit disk [9,10]. For more recent results on this class of operators, one may also consult the materials for instance in [12,13,16].

Various dynamical properties of the differentiation and Volterratype integral operators on generalized Fock spaces are studied. We show that the differentiation operator is always supercyclic on these spaces. We further characterize when it is hypercyclic, power bounded and uniformly mean ergodic. We prove that the operator satisfies the Ritt's resolvent condition if and only if it is power bounded and uniformly mean ergodic. Some similar results are obtained for the Volterra-type and Hardy integral operators.|f (z)| p e −pα|z| m dA(z) < ∞

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