We discuss boundedness and compactness of composition operators followed by multiplication as operators between Bloch-type spaces of analytic functions on the unit disk.
Ohno, Shûichi; Stroethoff, Karel; and Zhao, Ruhan, "Weighted Composition Operators between Bloch Type Spaces" (2003). Mathematics Faculty Publications. 1.
Ohno, S., Stroethoff, K. and Zhao, R. Weighted composition operators between Bloch type spaces, Rocky Mountain J. Math., 33(2003), 191--215.