Diagonalizing operators with reflection symmetry

Let $U$ be an operator in a Hilbert space $\mathcal{H}_{0}$, and let $\mathcal{K}\subset\mathcal{H}_{0}$ be a closed and invariant subspace. Suppose there is a period-2 unitary operator $J$ in $\mathcal{H}_{0}$ such that $JUJ=U^*$, and $PJP \geq 0$, where $P$ denotes the projection of $\mathcal{H}_{0}$ onto $\mathcal{K}$. We show that there is then a Hilbert space $\mathcal{H}(\mathcal{K})$, a contractive operator $W:\mathcal{K}\to\mathcal{H}(\mathcal{K})$, and a selfadjoint operator $S=S(U)$ in $\mathcal{H}(\mathcal{K})$ such that $W^*W=PJP$, $W$ has dense range, and $SW=WUP$. Moreover, given $(\mathcal{K},J)$ with the stated properties, the system $(\mathcal{H}(\mathcal{K}),W,S)$ is unique up to unitary equivalence, and subject to the three conditions in the conclusion. We also provide an operator-theoretic model of this structure where $U|_{\mathcal{K}}$ is a pure shift of infinite multiplicity, and where we show that $\ker(W)=0$. For that case, we describe the spectrum of the selfadjoint operator $S(U)$ in terms of structural properties of $U$. In the model, $U$ will be realized as a unitary scaling operator of the form $f(x)\mapsto f(cx)$, $c>1$, and the spectrum of $S(U_{c})$ is then computed in terms of the given number $c$.Comment: 30 pages; Dedicated to the memory of I.E. Sega

Discrete time quantum walks on percolation graphs

Randomly breaking connections in a graph alters its transport properties, a model used to describe percolation. In the case of quantum walks, dynamic percolation graphs represent a special type of imperfections, where the connections appear and disappear randomly in each step during the time evolution. The resulting open system dynamics is hard to treat numerically in general. We shortly review the literature on this problem. We then present our method to solve the evolution on finite percolation graphs in the long time limit, applying the asymptotic methods concerning random unitary maps. We work out the case of one dimensional chains in detail and provide a concrete, step by step numerical example in order to give more insight into the possible asymptotic behavior. The results about the case of the two-dimensional integer lattice are summarized, focusing on the Grover type coin operator.Comment: 22 pages, 3 figure

Classifying Higher Rank Toeplitz Operators.

To a higher rank directed graph (Λ, d), in the sense of Kumjian and Pask, 2000, one can associate natural noncommutative analytic Toeplitz algebras, both weakly closed and norm closed. We introduce methods for the classification of these algebras in the case of single vertex graphs