Let be the open disk of the complex plane. Then the Bloch space on and the little Bloch space on are defined by
where is the derivative of ;
With norm defined by , is a Banach space and is a closed subspace of . If is an analytic function on and , then the linear weighted composition operator is defined by
In the results of this paper, the authors derive characterizations for bounded and compact weighted composition operators. In particular, it is shown that
(1) is bounded on the Bloch space if and only if
(2) is compact on the Bloch space if and only if expressions of (1) converge to 0 as ; and is compact on if and only if the expressions of (1) converge to 0 as .
(3) is bounded on if the conditions of (1) are satisfied and as .