Substitution Vector-Valued Integral Type Operators on Orlicz Spaces

In this paper, we consider some fundamental properties of a substitution vector-valued integral operator Tu phi from Orlicz space L theta (mu) to Hilbert space H by the language of conditional expectation operators. First, we present necessary and sufficient conditions for boundedness and compactness Tu phi from L theta(mu) to H. Next, we investigate the problem of conditions on the generating Young functions, the functions u, phi and h = d(mu degrees phi-1)/d mu, under which operator Tu phi is of closed range or finite rank. Finally, we determine the lower and upper estimates for the essential norm of Tu phi on Orlicz spaces under certain conditions.