Compact Operators

If $\mathscr {X}$ is reflexive and $A \in \mathcal{B}(\mathscr {X}, \mathscr {Y})$, show that $A(\mathrm{ball}\ \mathscr {X})$ is closed in $\mathscr {Y}$.

Prove Proposition 3.5. Let $\mathscr {X}$, $\mathscr {Y}$, and $\mathscr {Z}$ be Banach spaces.

If $A \in \mathcal{B}_0(\mathscr {X}, \mathscr {Y})$, show that $\mathrm{cl}[\mathrm{ran}\ A]$ is separable.

If $A \in \mathcal{B}_0(\mathscr {X}, \mathscr {Y})$ and $\mathrm{ran}\ A$ is closed, show that $\mathrm{ran}\ A$ is finite dimensional.

If $A \in \mathcal{B}_0(\mathscr {X})$ and $A$ is invertible, show that $\dim \mathscr {X} < \infty$.

Let $(X, \Omega , \mu )$ be a finite measure space, $1 < p < \infty$, and $1/p + 1/q = 1$. If $k: X \times X \to \mathbb {F}$ is an $\Omega \times \Omega$-measurable function such that $\sup \left\{ \int \left|k(x, y)\right|^q\, d\mu (y): x \in X\right\} < \infty$, then $(Kf)(x) = \int k(x, y) f(y)\, d\mu (y)$ defines a compact operator on $L^p(\mu )$.

Let $(X, \Omega , \mu )$ be an arbitrary measure space, $1 < p < \infty$, and $1/p + 1/q = 1$. If $k: X \times X \to \mathbb {F}$ is an $\Omega \times \Omega$-measurable function such that

and if $(Kf)(x) = \int k(x, y) f(y)\, d\mu (y)$, then $K \in \mathcal{B}_0(L^p(\mu ))$ and $\left\| K\right\| \le M$.

Let $X$ be a compact space and let $\mu$ be a positive Borel measure on $X$. Let $T \in \mathcal{B}(L^p(\mu ), C(X))$ where $1 < p < \infty$. Show that if $A: L^p(\mu ) \to L^p(\mu )$ is defined by $Af = Tf$, then $A$ is compact.

(B.J. Pettis) If $\mathscr {X}$ is reflexive and $T \in \mathcal{B}(\mathscr {X}, l^1)$, then $T$ is a compact operator. Also, if $\mathscr {Y}$ is reflexive and $T \in \mathcal{B}(c_0, \mathscr {Y})$, $T$ is compact.

If $X$ is compact and $\left\{ f_1, \ldots , f_n, g_1, \ldots , g_n\right\} \subseteq C(X)$, define $k(x, y) = \sum_{j=1}^{n} f_j(x) g_j(y)$ for $x, y \in X$. Let $\mu$ be a regular Borel measure on $X$ and put $Kf(x) = \int k(x, y) f(y)\, d\mu (y)$. Show that $K \in \mathcal{B}(C(X))$ and $K$ has finite rank.

If $X$ is compact, $k \in C(X \times X)$, and $\mu$ is a regular Borel measure on $X$, show that $Kf(x) = \int k(x, y) f(y)\, d\mu (y)$ defines a compact operator on $C(X)$.

Let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space and for $\phi$ in $L^\infty (\mu )$ let $M_\phi : L^p(\mu ) \to L^p(\mu )$ be the multiplication operator defined in Example III.2.2. Give necessary and sufficient conditions on $(X, \Omega , \mu )$ and $\phi$ for $M_\phi$ to be compact.

Let $\tau : [0, 1] \to [0, 1]$ be continuous and define $A: C[0, 1] \to C[0, 1]$ by $Af = f \circ \tau$. Give necessary and sufficient conditions on $\tau$ for $A$ to be compact.

Let $A \in \mathcal{B}(c_0)$ and let $(\alpha_{mn})$ be the corresponding matrix as in Exercise 1.7. Give necessary and sufficient conditions on $(\alpha_{mn})$ for $A$ to be compact.

Let $A \in \mathcal{B}(l^1)$ and let $(\alpha_{mn})$ be the corresponding matrix as in Exercise 1.8. Give a necessary and sufficient condition on $(\alpha_{mn})$ for $A$ to be compact.

If $(X, d)$ is a compact metric space and $\mathscr {F} \subseteq C(X)$, show that $\mathscr {F}$ is equicontinuous if and only if for every $\varepsilon > 0$ there is a $\delta > 0$ such that $\left|f(x) - f(y)\right| < \varepsilon$ whenever $d(x, y) < \delta$ and $f \in \mathscr {F}$.

If $X$ is locally compact and $\mathscr {F} \subseteq C_0(X)$, show that $\mathscr {F}$ is totally bounded if and only if (a) $\mathscr {F}$ is bounded; (b) $\mathscr {F}$ is equicontinuous; (c) for every $\varepsilon > 0$ there is a compact subset $K$ of $X$ such that $\left|f(x)\right| < \varepsilon$ for all $f$ in $\mathscr {F}$ and $x$ in $X \setminus K$.

If $X$ is locally compact and $A \in \mathcal{B}_0(C_0(X))$, then there is a sequence $\left\{ A_n\right\}$ of finite-rank operators such that $\left\| A_n - A\right\| \to 0$.

Let $\mathscr {X}$ be a Banach space and suppose there is a net $\left\{ F_i\right\}$ of finite-rank operators on $\mathscr {X}$ such that (a) $\sup_i \left\| F_i\right\| < \infty$; (b) $\left\| F_i x - x\right\| \to 0$ for all $x$ in $\mathscr {X}$. Show that if $A \in \mathcal{B}_0(\mathscr {X})$, then $\left\| F_i A - A\right\| \to 0$ and hence there is a sequence $\left\{ A_n\right\}$ of finite-rank operators on $\mathscr {X}$ such that $\left\| A_n - A\right\| \to 0$.

Let $1 \le p \le \infty$ and let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space. If $A \in \mathcal{B}_0(L^p(\mu ))$, show that there is a sequence $\left\{ A_n\right\}$ of finite-rank operators such that $\left\| A_n - A\right\| \to 0$.

Let $X$ be compact and let $\mathscr {U}$ be the collection of all pairs $(C, F)$ where $C = \left\{ U_1, \ldots , U_n\right\}$ is a finite open cover of $X$ and $F = \left\{ x_1, \ldots , x_n\right\} \subseteq X$ such that $x_j \in U_j$ for $1 \le j \le n$. If $(C_1, F_1)$ and $(C_2, F_2) \in \mathscr {U}$, define $(C_1, F_1) \le (C_2, F_2)$ to mean:

1. $C_2$ is a refinement of $C_1$; that is, each member of $C_2$ is contained in some member of $C_1$.

2. $F_1 \subseteq F_2$. If $\alpha = (C, F) \in \mathscr {U}$ let $\left\{ \phi_1, \ldots , \phi_n\right\}$ be a partition of unity subordinate to $C$. If $F = \left\{ x_1, \ldots , x_n\right\}$, define $T_\alpha : C(X) \to C(X)$ by

Then: