\documentclass[minion]{homework}
\usepackage{cmacros}
\def\calB{\mathcal{B}}
\DeclareMathOperator{\Ker}{{\rm Ker}}
\DeclareMathOperator{\Image}{{\rm Image}}
\begin{document}
\doclabel{Math 617: Homework 10}
\docdate{Due: April 4, 2012}
\begin{aproblems}

\hproblem\solver{TJ Barry} Suppose $K\in L^2([0,1]\times[0,1])$. Define $T:L^2([0,1]\ra L^2([0,1]))$ by
$$
(Tf)x = \int_0^1 K(x,y)f(y)\; dy.
$$
Show that $T$ is compact.  Feel free to use the results of Examples
4.2.4 and 4.8.4 (which we effectively proved in class).

\hproblem Let $X$ be a Hilbert space.
We say a linear map $T$ is bounded below if there exists
a constant $c$ such that $||Tx||\ge c||x||$ for all $x\in X$.
Show that if $T:X\ra X$ is linear and continuous, it is
invertible if and only if it is bounded below and onto a dense subspace.

\hproblem\solver{Will Mitchell} Let $\{\lambda_k\}$ be a bounded sequence.  Define
$T:\ell_2\ra\ell_2$ by 
$$
T(x) = (\lambda_1 x_1,\lambda_2 x_2,\ldots).
$$
It is easy to see that $T$ is continuous; don't show this.  
Instead, determine $\sigma_p(T)$ and $\sigma(T)$.

\aproblem\solver{David Maxwell} D\&M 4.51

\end{aproblems}
\end{document}