The invariant subspace problem and Rosenblum operators I

Let $T\in B(\mathcal{H})$ be an invertible operator. From the 1940’s, Gelfand, Hille and Wermer investigated the invariant subspaces of $T$ by analyzing the growth of $\mathrm{\parallel }{T}^n\mathrm{\parallel }$, where $n\in \mathbb{Z}$. In this paper, we study the invariant subspaces of $T$ by estimating the growth of $\mathrm{\parallel }{T}^n+\lambda T^{-n}\mathrm{\parallel }$, where $n\in \mathbb{N}$ and $\lambda$ is a nonzero complex constant. The key ingredient of our approach is introducing the notion of shift representation operators, which is based on the Rosenblum operators. In addition, by employing shift representation operators, we provide an equivalent of the Invariant Subspace Problem via the injectivity of certain Hankel operators.