Functional analytic research of hyperbolic equations with memory

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A hyperbolic equation with memory and its related field are studied from the viewpoint of both sides of theory consideration to deal with the equation generally and systematically and the functional space in which the equation works place. As a result, abstract theories are developed to get global classical solutions for qualinear/semilinear evolution equations, and the characterizations are obtained of weighted composition operators on analytic function spaces and on the Bloch space. We also provided with the characterization of the contraction which corresponds injectively to the q-deformed normal operator.