\documentclass[12pt]{amsart}

\usepackage{amsmath,amssymb}

\title{Math Studies Homework 6}
\author{James Cummings}

\begin{document}

\maketitle


    Submit the \LaTeX\ file containing your solutions to the
    email address algebrahw@legba.math.cmu.edu by
    1159 pm on Friday 8 April.
 

\begin{enumerate}



\item   Let $V$ be a separable complex Hilbert space. Show that if $T \in B(V)$ and  
        $V$ has an ON Schauder basis consisting of eigenvectors of $T$, then
        $T T^\dagger = T^\dagger T$.  


\item   Let $V$ be a separable complex Hilbert space and let $A \in B(V)$ be self-adjoint.
        Show that $U = (A - i I)(A + i I)^{-1}$ is UNITARY.

\item   Let $V$ be a separable complex Hilbert space and let $M$ and $N$ be vector subspaces  
        such that $M + N = V$ and $M \perp N$. Show that $M$ and $N$ are closed.

\item   Let $V$ be a separable complex Hilbert space and let $T \in B(V)$. Prove that if
        $T^\dagger T$ is compact then $T$ is compact. 


\item  Let $(A_n)$ be an infinite sequence of $m \times m$ complex matrices. We will say that the
       sequence converges to a matrix $B$  if and only if the sequence  $(a^n_{i j})$ converges to $b_{i j}$ for all
       $i$ and $j$  with $1 \le i, j \le m$, where $a^n_{i, j}$ is the $(i, j)$ entry of $A_n$.
        
       Similarly an infinite sum 
       of $m \times m$ matrices $A_0 + A_1 + A_2 \ldots$ converges iff the sequence 
       of partial sums $A_0, A_0 + A_1, A_0 + A_1 + A_2, \ldots$  converges.

\begin{enumerate}

\item  Show that for any complex $m \times m$ matrix $A$ the ``exponential series'' 
\[
   exp(A) =   I + A + A^2/2! + A^3/3! + \ldots
\]
      converges.   Hint: let $\| A \|$ denote the max of the absolute values of the entries of $A$,
      and estimate $\| A^i \|$. 


\item  Derivatives of matrix valued functions are defined in the obvious (``entry by entry'') way.
       Show that the derivative of the matrix function $exp(t A)$ with respect to $t$ is
       $A exp(t A)$. 

\item  Show that if $A B = B A$ then $exp(A + B) = exp(A) exp(B)$. Is this true in general?
      
\item  Show that if the trace (sum of the diagonal entries) of $A$ is zero, then $exp(A)$ has
       determinant one. 

\item  Show that if $A$ is Hermitian then $exp(i A)$ is unitary. 


\end{enumerate}




\end{enumerate} 





\end{document}
