in this paper, we considered composition operators on weighted hilbert spaces of analytic functions and  observed that a formula for the  essential norm, gives a hilbert-schmidt characterization and characterizes the membership in schatten-class for these operators. also, closed range composition operators  are investigated.

For a complex polynomial or analytic function f , there is a strong correspondence between poles of the so-called local zeta functions or complex powers ∫ |f |2sω, where the ω are C∞ differential forms with compact support, and eigenvalues of the local monodromy of f . In particular Barlet showed that each monodromy eigenvalue of f is of the form exp(2π −1s0), where s0 is such a pole. We prove ...