Inner functions, invariant subspaces and cyclicity in $\mathcal{P}^t(\mu)$-spaces

We study the invariant subspaces generated by inner functions for a class of $\mathcal{P}^t(\mu)$-spaces which can be identified as spaces of analytic functions in the unit disk $\mathbb{D}$, where $\mu$ is a measure supported in the closed unit disk and $\mathcal{P}^t(\mu)$ is the span of analytic polynomials in the usual Lebesgue space $L^t(\mu)$. Our measures define a range of spaces somewhere in between the Hardy and the Bergman spaces, and our results are thus a mixture of results from these two theories. For a large class of measures $\mu$ we characterize the cyclic inner functions, and exhibit some interesting properties of invariant subspaces generated by non-cyclic inner functions. Our study is motivated by a connection with the problem of smooth approximations in de Branges-Rovnyak spaces.