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Homework 8, Real Analysis 2

Due April 20

Problem 1

Let $latex \{\phi_k\}_{1\le k<\infty}$ be a complete orthonormal system for $latex L^2(\R^d)$. Then $latex \{\phi_{k,j}\}_{1\le k,j < \infty}$ a complete orthonormal system for $latex L^2(\R^d\times\R^d)$, where $latex \phi_{k,j}(x,y) = \phi_k(x)\phi(y)$.

Problem 2

  1. Let P be the orthogonal projection onto a closed subspace S of a Hilbert space. Then $latex P^2 = P\text{ and } P^*=P$.

  2. Conversely, if P is a bounded operator such that $latex P^2 = P$ and $latex P^*=P$, then it is the orthogonal projection onto some closed subspace.

  3. Let $latex P_1\text{ and }P_2$ be the orthogonal projections onto the closed subspaces $latex S_1\text{ and }S_2$, respectively. Then $latex P_1P_2$ is an orthogonal projection if and only if they conmute, in that case, it projects onto $latex S_1\cap S_2$.

Problem 3

Let $latex \{u_k\}$ be a complete orthonormal system for a Hilbert space H, and $latex (a_k)$ a sequence of positive numbers such that $latex \sum_k a_k^2 < \infty$. Then

$latex \displaystyle A = \Big\{ \sum_k x_k u_k: |x_k|\le a_k \Big\}$

is compact in H.

Problem 4

Let T be a multiplier operator with multiplier sequence $latex (\lambda_k)$. Then T is compact if and only if $latex \lambda_k\to 0$.

Problem 5

  1. Let $latex T:L^2([0,1])\to L^2([0,1])$ be defined by $latex Tf(t) = tf(t)$. Then T is bounded and $latex T^*=T$, but is not compact nor has any eigenvectors.

  2. Let $latex T:H\to H$ be given, for a complete orthonormal system $latex \{u_k\}$, by $latex Tu_k = \dfrac{1}{k}u_{k+1}$. Then T is compact, but has no eigenvectors.