$\DeclareMathOperator{dist}{dist}\DeclareMathOperator{ker}{Ker}\DeclareMathOperator{im}{Im}$

1. Let $T\colon X → Y$ be a bounded linear operator between Hilbert spaces. Show the following equivalences:
   • $T$ is isometric$⇔⟨T x, T y⟩=⟨x, y⟩   ∀ x, y ∈ X ⇔ T^* T=I_X$.
   • $T$ is unitary$⇔ T^* T=I_X$ and $T T^*=I_Y$
2. Let $X$ be a Hilbert space and $U\colon X → X$ be a unitary operator.
   • Show that $\ker(I-U)=\ker(I-U^*)$;
   • Show that $X=\overline{\im(I-U)} ⊕ \ker(I-U)$;
   • Show that $\lim _{N → ∞} \frac{1}{N} \sum_{n=1}^{N-1} U^n x=x$ if $x ∈ \ker(I-U)$ and $\lim _{N → ∞} \frac{1}{N} \sum_{n=1}^{N-1} U^n x=0$ if $x ∈ \overline{\im(I-U)}$
   • Deduce that, for each $x ∈ X$, $$ \lim _{N → ∞} \frac{1}{N} \sum_{n=1}^{N-1} U^n x=P x, $$ where $P$ is the orthogonal projection onto $\ker(I-U)$.
3. Let $X$ be a real Hilbert space and assume that $A ∈ ℬ(X)$ is a projection, i.e. so that $A^2=A$. Show that $\im A=\ker(I-A)$ and prove the following equivalences $$ A=A^* ⟺(\im A)^⟂=\ker A ⟺{‖A‖}⩽1 . $$ Deduce that either ${‖A‖}=1$ or $A=0$ provided that one of the above statements is true. [Hint: To prove that ${‖A‖}⩽1$ implies $A=A^*$, show that $\dist(A x, \im(I-A))={‖Ax‖}$ and use this to prove that $⟨A x,(I-A) y⟩=0$ for all $x, y ∈ X$.]
4. Let $X=C([a, b])$, let $k:[a, b] ×[a, b] → ℝ$ be continuous. Show that $T ∈ ℬ(X)$ defined by $T f(x)=∫_a^b k(x, t) f(t) d t$ is a compact operator.
5. Let $X$ be a normed space, $Y$ a Banach space and let $(T_k) ⊂ ℬ(X, y)$ be a sequence of compact operators.
   • Let $(x_n)$ be a bounded sequence in $X$. Use a diagonal sequence arguement to show that there exists a subsequence $x_{n_j}$ so that $T_k(x_{n_j})$ converges for every $k ∈ ℕ$
   • Hence or otherwise show that if $T_k$ converges in $ℬ(X, Y)$ to an operator $T$ then $T$ is also a compact operator.