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We characterize the semigroups of composition operators that are stronglycontinuous on the mixed norm spaces $H(p,q,\alpha)$. First, we study theseparable spaces $H(p,q,\alpha)$ with $q<\infty,$ that behave as the Hardy andBergman spaces. In the second part we deal with the spaces$H(p,\infty,\alpha),$ where polynomials are not dense. To undertake this study,we introduce the integral operators, characterize its boundedness andcompactness, and use its properties to find the maximal closed linear subspaceof $H(p,\infty,\alpha)$ in which the semigroups are strongly continuous. Inparticular, we obtain that this maximal space is either $H(p,0,\alpha)$ ornon-separable, being this result the deepest one in the paper.

Semigroups of composition operators and integral operators on mixed norm spaces