The Diagonalization of Compact Self-Adjoint Operators

Prove Corollary 5.4: If $T$ is a compact self-adjoint operator, then there is a sequence $\left\{ \mu_{n}\right\}$ of real numbers and an orthonormal basis $\left\{ e_{n}\right\}$ for $(\ker T)^{\perp }$ such that for all $h$,

Note that there may be repetitions in the sequence $\left\{ \mu_{n}\right\}$. How many repetitions?

Prove Corollary 5.5: If $T \in \mathcal{B}_{0}(\mathcal{H})$, $T=T^{*}$, and $\ker T = (0)$, then $\mathcal{H}$ is separable.

Let $K$ and $k$ be as in Proposition 4.7 and suppose that $k(x,y) = \overline{k(y,x)}$. Show that $K$ is self-adjoint and if $\left\{ \mu_{n}\right\}$ are the eigenvalues of $K$, each repeated $\dim \ker (K-\mu_{n})$ times, then $\sum_{1}^{\infty } \left|\mu_{n}\right|^{2} < \infty$.

Proposition 4.7 (context): If $(X, \Omega , \mu )$ is a measure space and $k \in L^{2}(X \times X, \Omega \times \Omega , \mu \times \mu )$, then

is a compact operator and $\left\| K\right\| \leq \left\| k\right\|_{2}$.

If $T$ is a compact self-adjoint operator and $\left\{ e_{n}\right\}$ and $\left\{ \mu_{n}\right\}$ are as in (5.4) and if $h$ is a given vector in $\mathcal{H}$, show that there is a vector $f$ in $\mathcal{H}$ such that $Tf=h$ if and only if $h \perp \ker T$ and $\sum_{n} \mu_{n}^{-2} \left| \left\langle h, e_{n} \right\rangle \right|^{2} < \infty$. Find the form of the general vector $f$ such that $Tf=h$.

(5.4) (context): There is a sequence $\left\{ \mu_{n}\right\}$ of real numbers and an orthonormal basis $\left\{ e_{n}\right\}$ for $(\ker T)^{\perp }$ such that for all $h$, $Th = \sum_{n=1}^{\infty } \mu_{n} \left\langle h, e_{n} \right\rangle e_{n}$.

Let $T$, $\left\{ \mu_{n}\right\}$, and $\left\{ e_{n}\right\}$ be as in (5.4). If $\lambda \neq 0$ and $\lambda \neq \mu_{n}$ for any $\mu_{n}$, then for every $h$ in $\mathcal{H}$ there is a unique $f$ in $\mathcal{H}$ such that $(\lambda - T)f = h$. Moreover,

Interpret this when $T$ is an integral operator.

(5.4) (context): There is a sequence $\left\{ \mu_{n}\right\}$ of real numbers and an orthonormal basis $\left\{ e_{n}\right\}$ for $(\ker T)^{\perp }$ such that for all $h$, $Th = \sum_{n=1}^{\infty } \mu_{n} \left\langle h, e_{n} \right\rangle e_{n}$.