L¹ factorizations and invariant subspaces for weighted composition operators

Abstract.

For a contraction operator T with spectral radius less than one on a Banach space $\user1{\mathcal{X}}$, it is shown that the factorization of certain L¹ functions by vectors x in $\user1{\mathcal{X}}$ and x*. in $\user1{\mathcal{X}}*$, in the sense that $\left\langle {T^n x,x* } \right\rangle = \widehat{f}( - n)$ for n ≧ 0, implies the existence of invariant subspaces for T. Explicit formulae for such factorizations are given in the case of weighted composition operators on reproducing kernel Hilbert spaces. An interpolation result of McPhail is applied to show how this can be used to construct invariant subspaces of hyperbolic weighted composition operators on H².