Odometer maps on Fock spaces: block decompositions, Toeplitz-type realizations, and the adjoint

Abstract

We study odometer maps $W_L$ on vector-valued full Fock spaces arising from Fock representations of the odometer semigroup. We obtain a canonical upper triangular block decomposition \[ W_L= \begin{pmatrix} W_{11} & W_{12}\\ 0 & W_{22} \end{pmatrix}, \] where $W_{11}$ is unitary and $W_{22}$ admits a Hardy space realization as an analytic Toeplitz operator $M_\Theta$. The associated symbol $\Theta\in H^\infty_{\mathcal{B}(\mathcal{E})}(\mathbb{D})$ is used to characterize the isometric, unitary, and invertible cases, as well as norm identities and Douglas-type factorization properties of $W_L$. We also derive an explicit formula for $W_L^*$ for arbitrary bounded symbols $L$. In the isometric case, this identifies $\ker W_L^*$ with $\mathcal{E}_L\ominus L\mathcal{E}$, and hence $\operatorname{mult}(W_L)=\dim(\mathcal{E}_L\ominus L\mathcal{E})=\operatorname{mult}(M_\Theta)$. In the same setting, the condition $\dim(\mathcal{E}_L\ominus L\mathcal{E})<\infty$ is equivalent both to Fredholmness and to essential normality of $W_L$, with $\operatorname{ind}(W_L)=-\dim(\mathcal{E}_L\ominus L\mathcal{E})$. We further obtain Coburn-type spectral consequences and a necessary condition for hyponormality.