Linear Operators on a Banach Space

Prove Proposition 1.3: if $\mathscr {X}$ and $\mathscr {Y}$ are Banach spaces, $A, B \in \mathcal{B}(\mathscr {X}, \mathscr {Y})$, and $\alpha , \beta \in \mathbb {F}$, then $(\alpha A + \beta B)^* = \alpha A^* + \beta B^*$. Moreover, $A^*: (\mathscr {Y}^*, \mathrm{wk}^*) \to (\mathscr {X}^*, \mathrm{wk}^*)$ is continuous.

Complete the proof of Proposition 1.4: if $\mathscr {X}$ and $\mathscr {Y}$ are Banach spaces and $A \in \mathcal{B}(\mathscr {X}, \mathscr {Y})$, then:

Verify the statement made in (1.5): let $(X, \Omega , \mu )$ and $M_\phi : L^p(\mu ) \to L^p(\mu )$ be as in Example III.2.2. If $1 \le p < \infty$ and $1/p + 1/q = 1$, then $M_\phi^*: L^q(\mu ) \to L^q(\mu )$ is given by $M_\phi^* f = \phi f$. That is, $M_\phi^* = M_\phi$.

Verify the statement made in (1.6): let $K$ and $k$ be as in Example III.2.3. If $1 \le p < \infty$ and $1/p + 1/q = 1$, then $K^*: L^q(\mu ) \to L^q(\mu )$ is the integral operator with kernel $k^*(x, y) \equiv k(y, x)$.

Verify the statement made in (1.7): let $X$, $Y$, $\tau$, and $A$ be as in Example III.2.4. Then $A^*: M(Y) \to M(X)$ is given by

for every Borel subset $\Delta$ of $X$ and every $\mu$ in $M(Y)$.

Let $1 \le p < \infty$ and define $S: l^p \to l^p$ by $S(\alpha_1, \alpha_2, \ldots ) = (0, \alpha_1, \alpha_2, \ldots )$. Compute $S^*$.

Let $A \in \mathcal{B}(c_0)$ and for $n \ge 1$, define $e_n$ in $c_0$ by $e_n(n) = 1$ and $e_n(m) = 0$ for $m \ne n$. Put $\alpha_{mn} = (Ae_n)(m)$ for $m, n \ge 1$.

Let $A \in \mathcal{B}(l^1)$ and for $n \ge 1$ define $e_n$ in $l^1$ by $e_n(n) = 1$, $e_n(m) = 0$ for $m \ne n$. Put $\alpha_{mn} = (Ae_n)(m)$ for $m, n \ge 1$.

(Bonsall [1986]) Let $\mathscr {X}$ be a Banach space, $Z$ a nonempty set, and $u: Z \to \mathscr {X}$. If there are positive constants $M_1$ and $M_2$ such that (i) $\left\| u(z)\right\| \le M_1$ for all $z$ in $Z$ and (ii) for every $x^*$ in $\mathscr {X}$, $\sup \left\{ \left|\langle u(z), x^* \rangle \right|: z \in Z\right\} \ge M_2 \left\| x^*\right\|$; then for every $x$ in $\mathscr {X}$ there is an $f$ in $l^1(Z)$ such that $(*)\ x = \sum \left\{ f(z)u(z): z \in Z\right\}$ and $M_2 \inf \left\| f\right\|_1 \le \left\| x\right\| \le M_1 \inf \left\| f\right\|_1$, where the infimum is taken over all $f$ in $l^1(Z)$ such that $(*)$ holds.

(Bonsall [1986]) Let $m$ be normalized Lebesgue measure on $\partial \mathbb {D}$ and for $\left|z\right| < 1$ and $\left|w\right| = 1$ let $p_z(w) = (1 - \left|z\right|^2)/\left|1 - \bar{z}w\right|^2$. So $p_z$ is the Poisson kernel. Show that if $f \in L^1(m)$, then there is a sequence $\left\{ z_n\right\} \subseteq \mathbb {D}$ and a sequence $\left\{ \lambda_n\right\}$ in $l^1$ such that $(*)\ f = \sum_{n=1}^{\infty } \lambda_n p_{z_n}$. Moreover, $\left\| f\right\|_1 = \inf \sum_{n=1}^{\infty } \left|\lambda_n\right|$, where the infimum is taken over all $\left\{ \lambda_n\right\}$ in $l^1$ such that $(*)$ holds.

If $\mathscr {X}$ and $\mathscr {Y}$ are Banach spaces and $B \in \mathcal{B}(\mathscr {Y}^*, \mathscr {X}^*)$, then there is an operator $A$ in $\mathcal{B}(\mathscr {X}, \mathscr {Y})$ such that $B = A^*$ if and only if $B$ is $\mathrm{wk}^*$-continuous.

If $\mathscr {X}$ is a Banach space and $\mathscr {M}$ and $\mathscr {N}$ are closed subspaces, show that the following statements are equivalent.

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:

Let $A, B, T \in \mathcal{B}(\mathscr {X})$ such that $TA = BT$. Show that $\mathrm{graph}(T) \in \mathrm{Lat}(A \oplus B)$.

Prove that $\mathscr {M} \in \mathrm{Lat}\ T$ if and only if $\mathscr {M}^\perp \in \mathrm{Lat}\ T^*$. What does the map $\mathscr {M} \mapsto \mathscr {M}^\perp$ of $\mathrm{Lat}\ T$ into $\mathrm{Lat}\ T^*$ do to the lattice operations?

Let $\left\{ e_1, e_2, e_3\right\}$ be the usual basis for $\mathbb {F}^3$ and let $\alpha_1, \alpha_2, \alpha_3 \in \mathbb {F}$. Define $T: \mathbb {F}^3 \to \mathbb {F}^3$ by $Te_j = \alpha_j e_j$, $1 \le j \le 3$.

Generalize Exercise 3 by characterizing $\mathrm{Lat}\ T$, where $T$ is defined by $Te_j = \alpha_j e_j$, $1 \le j \le d$, for any choice of scalars $\alpha_1, \ldots , \alpha_d$ and where $\left\{ e_1, \ldots , e_d\right\}$ is the usual basis for $\mathbb {F}^d$.

Let $\left\{ e_1, \ldots , e_d\right\}$ be the usual basis for $\mathbb {F}^d$, let $\left\{ \alpha_1, \ldots , \alpha_{d-1}\right\} \subseteq \mathbb {F}$ with no $\alpha_j = 0$. If $Te_j = \alpha_j e_{j+1}$ for $1 \le j \le d - 1$ and $Te_d = 0$, find $\mathrm{Lat}\ T$.

If $T \in \mathcal{B}(\mathbb {R}^d)$ and $d \ge 3$, show that $T$ has a nontrivial subspace.

Show that if $T \in \mathcal{B}(\mathscr {X})$ and $\mathscr {X}$ is not separable, then $T$ has a nontrivial invariant subspace.

Give an example of an invertible operator $T$ on a Banach space $\mathscr {X}$ and an invariant subspace $\mathscr {M}$ for $T$ such that $\mathscr {M}$ is not invariant for $T^{-1}$.

Let $\mathscr {X}$ be a Banach space over $\mathbb {C}$, let $K \in \mathcal{B}_0(\mathscr {X})$ and show that if $\mathscr {C}$ is a maximal chain in $\mathrm{Lat}\ K$, then $\mathscr {C}$ is a maximal chain in the lattice of all subspaces of $\mathscr {X}$.

Prove Proposition 5.2.

If $\mathscr {R}$ and $\mathscr {X}$ are Banach spaces and $\Phi : \mathscr {R} \to \mathscr {X}$ is an isometry, give an elementary proof that $\Phi^*$ is surjective.

Let $\mathscr {X}$ be a Banach space and recall the definition of a weakly Cauchy sequence (V.4.4).

Say that a Banach space $\mathscr {X}$ is weakly compactly generated (WCG) if there is a weakly compact subset $K$ of $\mathscr {X}$ such that $\mathscr {X}$ is the closed linear span of $K$. Prove (Davis, Figiel, Johnson, and Pelczynski, [1974]) that $\mathscr {X}$ is WCG if and only if there is a reflexive space and an injective bounded operator $T: \mathscr {R} \to \mathscr {X}$ such that $\mathrm{ran}\ T$ is dense.

If $(X, \Omega , \mu )$ is a finite measure space, $k \in L^\infty (X \times X, \Omega \times \Omega , \mu \times \mu )$, and $K: L^1(\mu ) \to L^1(\mu )$ is defined by $(Kf)(x) = \int k(x, y) f(y)\, d\mu (y)$, show that $K$ is weakly compact and $K^2$ is compact.

Let $\mathscr {Y}$ be a weakly sequentially complete Banach space. That is, if $\left\{ y_n\right\}$ is a sequence in $\mathscr {Y}$ such that $\left\{ \langle y_n, y^* \rangle \right\}$ is a Cauchy sequence in $\mathbb {F}$ for every $y^*$ in $\mathscr {Y}^*$, then there is a $y$ in $\mathscr {Y}$ such that $y_n \to y$ weakly.