# Group Work 2
###### tags: `224a`
I. Consider the operator $(A\psi)_n = \psi_{n-1}+\psi_{n+1}$ on $\ell^2(\mathbb{Z})$ from the previous lecture. We showed via Fourier series that $A$ is unitarily equivalent to $M_{2\cos(x)}$ on $L^2([0,2\pi),dx/2\pi)$ and has spectrum $[-2,2]$.

0. Find another function $g$ and measure space $(X,\nu)$ such that $A$ is unitarily equivalent to $M_g$ on $L^2(X,\nu)$

Let $(Bf)(x)=xf(x)$ on $L^2([-2,2],dx)$.
1. Calculate the point and continuous spectrum of $A$ and $B$.
2. Show that $\|f(A)\|=\|f(B)\|$ for every $f\in C([-2,2])$.
4. Show that $A$ is unitarily equivalent to $B\oplus B$, i.e., there is a decomposition $L^2([-2,2],dx)=H_1\oplus H_2$ and an isometry $U$ such that $UAU^{-1}=B\oplus B$ on $H_1\oplus H_2$. Find the isometry.
4. Does $A$ have a cyclic vector?

II. Let $T_d$ denote the infinite $d-$regular tree with a distinguished root vertex $r$. Let $(Cf)(x) = \sum_{y\sim x} f(y)$ be the adjacency operator of $T_d$ on $\ell^2(T_d)$. Calculate the point, continuous, and residual spectrum of $T_d$.

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Group Summaries:

1.Dipti Jasrasaria, Nicole Farias, Samuel Olivier, Peter Sokurov, Mathias Palmstroem, Sarvesh Sadana: We spent about half the time showing that the adjacency operator in fourier space is a multiplication operator. We discussed what a measure space actually is but were confused about how to choose both a function g and a new measure space. We were able to show the spectrum for I.1 is [-2,2].
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