# How may a multiplication operator change the spectrum of a random walk

The motivation for this question comes from similar questions about the horocyclic flow and how it is perturbed by some multiplications operator. Since the random walk operator is also broadly studied, I was wondering what are the references/results/conjectures/problems/examples in this setting. So, though I tried to make the question slightly broader, if it helps, take $B$ below to be a simple random walk operator.

Consider an bounded operator $B:\ell^2(\mathbb{N}) \to \ell^2(\mathbb{N})$ and a multiplication operator $\phi_f$. My first naive question would be what can one expect of the spectrum of $\phi_f \circ B$ assuming the spectrum of $B$ is known. Since the answer seems to be "not much", here are a few extra restrictions:

$\mathbf{Question}$: Let $B:\ell^2(\mathbb{N}) \to \ell^2(\mathbb{N})$ be a bounded self-adjoint operator. Take $f:\mathbb{N} \to \mathbb{N}$. Assume there exists a $N \in \mathbb{N}$ with

a) $1\leq f(n) \leq N$ i.e. $f: \mathbb{N} \to \lbrace 1,2, \ldots, N \rbrace$

b) $B$ is of the type $B e_n = a_n \chi_{S_n} + b_n \chi_{T_n}$ where $\chi_{S}$ is the characteristic function of $S \subset \mathbb{N}$, $|S_n| \leq N$, $|T_n| \leq N$ and $a_n,b_n \in \lbrace 1,2, \ldots, N \rbrace$.

Let $\phi_f$ be the multiplication operator associated to $f$ (i.e. $(\phi_f g)(n) = f(n)g(n)$ ). Assume the spectrum of $B$ is absolutely continuous (with respect to the Lebesgue measure). Is the spectrum of $\phi_f \circ B$ absolutely continuous?

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