Let $cal{B}$ be a reflexive Banach space of functions analytic on the open unit disc and M be an invariant subspace of the multiplication operator by the independent variable, $M_z$. Suppose that $varphi;in;cal{H}^{infty}$ and $M_{varphi}$ : M ${ ightarrow}$ M, defined by $M_{varphi}f={varphi}f$, is the operator of multiplication by ${varphi}$. We would like to investigate the spectrum and the essential spectrum of $M_{varphi}$ and we are looking for the necessary and sufficient conditions for $M_{varphi}$ to be a Fredholm operator. Also we give a sufficient condition for a sequence ${w_n}$ to be an interpolating sequence for $cal{B}$. At last the commutant of $M_{varphi}$ under certain conditions on M and ${varphi}$ is determined.