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
   $\Vert f(A)\Vert =\Vert f(B)\Vert$ for every
   $f\in C([-2,2])$.
3. 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
   $UA{U}^{-1}=B\oplus B$ 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

$T_d$.

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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