Ir al contenido principal

## Due April 20

### Problem 1

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

### Problem 2

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

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

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

### Problem 3

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

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

is compact in H.

### Problem 4

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

### Problem 5

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

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