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\title{Math Studies Homework 5}
\author{James Cummings}

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    Submit the \LaTeX\ file containing your solutions to the
    email address algebrahw@legba.math.cmu.edu by
    1159 pm on Friday 25 March.
 

    A point of clarification: when we say that an operator $K$ in
    $B(V)$ is invertible we mean that there is $L$ in $B(V)$
    such that $KL= LK = I$ where as usual $I$ is the identity.
    
    As we see in class soon, if a map in $B(V)$ is bijective then
    its inverse is bounded. So a non-invertible $K$ in $B(V)$ fails
    to be bijective, hence either it's not into or not onto.
    If V is FD then not-into equals not-onto but they are distinct
    in $l_2$; consider the maps taking $(x_0, x_1, x_2 \ldots)$
    to  $(0, x_0, x_1, \ldots)$ and $(x_1, x_2, ldots)$.

\begin{enumerate}


\item   Let $H$ be the subset of $l_2(\Bbb C)$ consisting of those
    sequences with $\vert x_n \vert \le 1/(n+1)$. Show that
    $H$ is compact. Hint: sequential compactness.

\item   Let $K$ be the bounded operator on $l_2(\Bbb C)$ 
      which maps $(x_n)$ to $(x_n/(n+1))$. Show that
      $K$ is compact.  (Trickier) For which sequences of complex numbers
      $(\alpha_n)$ is the map taking $(x_n)$ to
      $(\alpha_n x_n)$ compact? 

\item Let $K$ be as in the previous question.
      Find the eigenvalues of $K$ and show that $0$ is in the spectrum
      but not an eigenvalue. Is $0$ the only point in the spectrum which is
      not an eigenvalue?

\item Let $V = C^n$ and let $T$ be a linear operator on $V$. Show that
      $T T^\dagger$ is self-adjoint. Are all self-adjoint operators of this
      form?

\item  Let $V$ and $W$ be complex Hilbert spaces. Show how to give $V \oplus  W$
       a Hilbert space structure so that $V \oplus \{ 0 \}$ is 
       a closed subspace IMic to $V$ and $\{ 0 \} \oplus W$
       is a closed subspace IMIc to $W$. 
       Can you find a reasonable Hilbert space structure for $V \otimes W$?




\end{enumerate}



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