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 \(f\in C([-2,2])\).
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 \(UAU^{-1}=B\oplus B\) 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 \(T_d\).

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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