Banach Algebras and Spectral Theory for Operators on a Banach Space

Let $\mathcal{A}$ be an algebra that is also a Banach space and such that if $a \in \mathcal{A}$, the maps $x \mapsto ax$ and $x \mapsto xa$ of $\mathcal{A} \to \mathcal{A}$ are continuous. Let $\mathcal{A}_1 = \mathcal{A} \times \mathbb {F}$ as in Proposition 1.3. If $a \in \mathcal{A}$, define $L_a: \mathcal{A}_1 \to \mathcal{A}_1$ by $L_a(x, \xi ) = (ax + \xi a, 0)$. Show that $L_a \in \mathcal{B}(\mathcal{A}_1)$ and if $\vertiii {a} = \left\| L_a\right\|$, then $\vertiii {\cdot }$ is equivalent to the norm of $\mathcal{A}$ and $\mathcal{A}$ with $\vertiii {\cdot }$ is a Banach algebra.

Complete the proof of the following proposition. If $\mathcal{A}$ is a Banach algebra without an identity, let $\mathcal{A}_1 = \mathcal{A} \times \mathbb {F}$. Define algebraic operations on $\mathcal{A}_1$ by

Define $\left\| (a, \alpha )\right\| = \left\| a\right\| + \left|\alpha \right|$. Then $\mathcal{A}_1$ with this norm and the algebraic operations defined above is a Banach algebra with identity $(0, 1)$ and $a \mapsto (a, 0)$ is an isometric isomorphism of $\mathcal{A}$ into $\mathcal{A}_1$.

Verify the statements made in Examples (1.4) through (1.9) and (1.11).

Let $G$ be a locally compact group.

Prove the following proposition. If $\{ \mathcal{A}_i\}$ is a collection of Banach algebras, then $\bigoplus_0 \mathcal{A}_i$ and $\bigoplus_\infty \mathcal{A}_i$ are Banach algebras, where $\bigoplus_0 \mathcal{A}_i \equiv \left\{ a \in \prod_i \mathcal{A}_i : \text{for all } \varepsilon > 0, \{ i : \left\| a(i)\right\| \ge \varepsilon \} \text{ is finite}\right\}$.

Let $\{ \mathcal{A}_i : i \in I\}$ be a collection of Banach algebras.

If $X, Y$ are completely regular, show that $C_b(X) \oplus_\infty C_b(Y)$ is isometrically isomorphic to $C_b(X \oplus Y)$, where $X \oplus Y$ is the disjoint union of $X$ and $Y$.

If $X$ and $Y$ are locally compact, show that $C_0(X) \oplus_\infty C_0(Y)$ is isometrically isomorphic to $C_0(X \oplus Y)$.

Let $\{ X_i : i \in I\}$ be a collection of locally compact spaces and let $X =$ the disjoint union of these spaces furnished with the topology $\left\{ U \subseteq X : U \cap X_i \text{ is open in } X_i \text{ for all } i\right\}$. Show that $X$ is locally compact and $\bigoplus_0 C_0(X_i)$ is isometrically isomorphic to $C_0(X)$.

Let $\mathcal{A}$ be a Banach algebra and let $\mathcal{L}$ be all of the closed left ideals in $\mathcal{A}$. If $I_1, I_2 \in \mathcal{L}$, define $I_1 \vee I_2 \equiv \text{cl}(I_1 + I_2)$ and $I_1 \wedge I_2 = I_1 \cap I_2$. Show that with these definitions $\mathcal{L}$ is a complete lattice with a largest and a smallest element.

Let $X$ be locally compact. For every open subset $U$ of $X$, let $I(U) = \left\{ \phi \in C_0(X) : \phi = 0 \text{ on } X \setminus U\right\}$. Show that $U \mapsto I(U)$ is a lattice monomorphism of the collection of open subsets of $X$ into the lattice of closed ideals of $C_0(X)$. (It is, in fact, surjective, but the proof of that should wait.)

Let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space and let $I$ be an ideal in $L^\infty (X, \Omega , \mu )$ that is weak* closed. Show that there is a set $\Delta$ in $\Omega$ such that $I = \left\{ \phi \in L^\infty (X, \Omega , \mu ) : \phi = 0 \text{ a.e. on } \Delta \right\}$.

Let

and let

Show that $\mathcal{A}$ is a Banach algebra and $\mathcal{M}$ is a maximal ideal in $\mathcal{A}$.

Show that for $n \ge 1$, $M_n(\mathbb {C})$ has no nontrivial ideals. How about $M_n(\mathbb {R})$?

Let $\mathcal{A}$ be a Banach algebra but do not assume that $\mathcal{A}$ has an identity. If $I$ is a left ideal of $\mathcal{A}$, say that $I$ is a modular left ideal if there is a $u$ in $\mathcal{A}$ such that $\mathcal{A}(1 - u) \equiv \left\{ a - au : a \in \mathcal{A}\right\} \subseteq I$; call such an element $u$ of $\mathcal{A}$ a right modular unit for $I$. Similarly, define right modular ideals and left modular units. Prove the following.

Using the terminology of Exercise 6, let $I$ be an ideal of $\mathcal{A}$. Show:

If $\mathcal{A}$ is a Banach algebra, a net $\{ e_i\}$ in $\mathcal{A}$ is called an approximate identity for $\mathcal{A}$ if $\sup_i \left\| e_i\right\| < \infty$ and for each $a$ in $\mathcal{A}$, $e_i a \to a$ and $a e_i \to a$. Show that $\mathcal{A}$ has an approximate identity if and only if there is a bounded subset $E$ of $\mathcal{A}$ such that for every $\varepsilon > 0$ and for every $a$ in $\mathcal{A}$ there is an $e$ in $E$ with $\left\| ae - a\right\| + \left\| ea - a\right\| < \varepsilon$.

Show that if $X$ is locally compact, then $C_0(X)$ has an approximate identity.

If $\mathcal{H}$ is a Hilbert space, show that $\mathcal{B}_0(\mathcal{H})$ has an approximate identity.

If $G$ is a locally compact group, show that $L^1(G)$ (1.11) has an approximate identity.

For $0 < r < 1$, let $P_r: \partial \mathbb {D} \to [0, \infty )$ be defined by $P_r(z) = \sum_{n=-\infty }^\infty r^{\left|n\right|} z^n$ (the Poisson kernel). Show that $\{ P_r\}$ is an approximate identity for $L^1(\partial \mathbb {D})$ (under convolution).

If $\mathcal{H}$ is a Hilbert space and $P$ is a projection, show that $\mathcal{B}_0(\mathcal{H})P$ is a closed modular left ideal of $\mathcal{B}_0(\mathcal{H})$. What is the associated right modular unit?

Find the minimal proper left ideals of $M_n(\mathbb {F})$.

Find the minimal closed proper left ideals of $\mathcal{B}_0(\mathcal{H})$, $\mathcal{H}$ a Hilbert space. How about for $\mathcal{B}_0(\mathcal{X})$, $\mathcal{X}$ a Banach space?

What are the maximal modular left ideals of $\mathcal{B}_0(\mathcal{H})$, $\mathcal{H}$ a Hilbert space?

Let $S$ be the unilateral shift on $\ell^2$, the Hilbert space operator defined by $S(x_1, x_2, x_3, \ldots ) = (0, x_1, x_2, \ldots )$. Show that $S$ is left invertible but not right invertible.

If $\mathcal{A}$ is a Banach algebra with identity and $a \in \mathcal{A}$ and is nilpotent (that is, $a^n = 0$ for some $n$), then $\sigma (a) = \{ 0\}$.

Let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space and let $\mathcal{A} = L^\infty (X, \Omega , \mu )$ (1.6). If $\phi \in \mathcal{A}$, show that the following are equivalent:

If $G$ is an open subset of $\mathbb {C}$ and $f: G \to \mathcal{X}$ is a function such that for each $x^*$ in $\mathcal{X}^*$, $x^* \circ f: G \to \mathbb {C}$ is analytic, then $f$ is analytic. If the word continuous'' is substituted for both occurrences of the word analytic,'' is the preceding statement still true?

If $\mathcal{A}$ is a Banach algebra with identity, $\{ a_n\} \subseteq \mathcal{A}$, $a_n \to a$, $\alpha_n \in \sigma (a_n)$, and $\alpha_n \to \alpha$, then $\alpha \in \sigma (a)$.

If $\mathcal{A}$ is a Banach algebra with identity and $r: \mathcal{A} \to [0, \infty )$ is the spectral radius, show that $r$ is upper semicontinuous. If $a \in \mathcal{A}$ such that $r(a) = 0$, show that $r$ is continuous at $a$.

If $\mathcal{A}$ is a Banach algebra with identity, $a, b \in \mathcal{A}$, and $\alpha$ is a nonzero scalar such that $(\alpha - ab)$ is invertible, show that $(\alpha - ba)$ is invertible and $(\alpha - ba)^{-1} = \alpha^{-1} + \alpha^{-1} b(\alpha - ab)^{-1} a$. Show that $\sigma (ab) \cup \{ 0\} = \sigma (ba) \cup \{ 0\}$ and give an example such that $\sigma (ab) \ne \sigma (ba)$.

Let $\mathcal{A} = C(X)$, $X$ compact (see Example 3.2). If $g \in C(X)$ and $f \in \text{Hol}(g)$, show that $f(g) = f \circ g$.

Let $a$ be a nilpotent element of $\mathcal{A}$. For $f, g$ in $\text{Hol}(a)$, give a necessary and sufficient condition on $f$ and $g$ that $f(a) = g(a)$.

Let $d \ge 1$ and let $A \in M_d(\mathbb {C})$. Give a necessary and sufficient condition on $f$ in $\text{Hol}(A)$ such that $f(A) = 0$.

If $\mathcal{A}$ is a Banach algebra with identity, $a \in \mathcal{A}$, $f \in \text{Hol}(a)$, and $g$ is analytic in a neighborhood of $f(\sigma (a))$, then $g \circ f \in \text{Hol}(a)$ and $g(f(a)) = g \circ f(a)$.

If $\mathcal{X}$ is a Banach space, $A \in \mathcal{B}(\mathcal{X})$, and $\mathcal{M} \le \mathcal{X}$ such that $(A - \alpha )^{-1} \mathcal{M} \subseteq \mathcal{M}$ for all $\alpha$ in $\rho (A)$, show that $f(A) \mathcal{M} \subseteq \mathcal{M}$ whenever $f \in \text{Hol}(A)$.

If $\mathcal{X}$ is a Banach space, $A \in \mathcal{B}(\mathcal{X})$, and $f \in \text{Hol}(A)$, show that $f(A)^* = f(A^*)$. (See (6.1) below.)

If $\mathcal{H}$ is a Hilbert space, $A \in \mathcal{B}(\mathcal{H})$, and $f \in \text{Hol}(A)$, show that $f(A)^* = \tilde{f}(A^*)$, where $\tilde{f}(z) = \overline{f(\bar{z})}$. (See (6.1) below.)

If $\mathcal{H}$ is a Hilbert space, $A$ is a normal operator on $\mathcal{H}$, and $f \in \text{Hol}(A)$, show that $f(A)$ is normal.

Let $\mathcal{X}$ be a Banach space and let $A \in \mathcal{B}(\mathcal{X})$. Show that if $\sigma (A) = F_1 \cup F_2$ where $F_1, F_2$ are disjoint closed subsets of $\mathbb {C}$, then there are topologically complementary subspaces $\mathcal{X}_1, \mathcal{X}_2$ of $\mathcal{X}$ such that

Let $A \in M_d(\mathbb {C})$, $\sigma (A) = \{ \alpha_1, \ldots , \alpha_n\}$, where $\alpha_i \ne \alpha_j$ for $i \ne j$. Show that for $1 \le j \le n$ there is a matrix $A_j$ in $M_{d_j}(\mathbb {C})$ such that $\sigma (A_j) = \{ \alpha_j\}$ and $A$ is similar to $A_1 \oplus \cdots \oplus A_n$.

If $\mathcal{A}$ is a Banach algebra, $I$ is an ideal of $\mathcal{A}$ (not necessarily closed), $a \in I$, and $f \in \text{Hol}(a)$ such that $f(0) = 0$, show that $f(a) \in I$.

If $K$ is a compact subset of $\mathbb {C}$, let $P(K)$ be the closure of the polynomials in $C(K)$. Show that the identity map on polynomials extends to an isometric isomorphism of $P(K)$ onto $P(\hat{K})$.

If $K$ is a compact subset of $\mathbb {C}$, let $R(K)$ be the closure in $C(K)$ of all rational functions with poles off $K$. If $f \in R(K)$, show that $\sigma_{R(K)}(f) = f(K)$. If $f \in P(K)$, show that $\sigma_{P(K)}(f) = \hat{f}(\hat{K})$, where $\hat{f}$ is a natural extension of $f$ to $\hat{K}$.

Let $\mathcal{A}$ and $\mathcal{B}$ be Banach algebras with a common identity such that $\mathcal{B} \subseteq \mathcal{A}$ and $a \in \mathcal{B}$; recall that then $\sigma_{\mathcal{A}}(a) \subseteq \sigma_{\mathcal{B}}(a)$ and $\sigma_{\mathcal{A}}(a)$ and $\sigma_{\mathcal{B}}(a)$ have the same polynomially convex hull. If $\sigma_{\mathcal{A}}(a) \subseteq \mathbb {R}$, show that $\sigma_{\mathcal{B}}(a) = \sigma_{\mathcal{A}}(a)$.

Let $\mathcal{A}$ be a Banach algebra with identity and let $a \in \mathcal{A}$. If $G_1, G_2, \ldots$ are the holes of $\sigma_{\mathcal{A}}(a)$ and $1 \le n_1 \le n_2 \le \cdots$, show that there is a subalgebra $\mathcal{B}$ of $\mathcal{A}$ such that $a \in \mathcal{B}$ and $\sigma_{\mathcal{B}}(a) = \sigma_{\mathcal{A}}(a) \cup \bigcup_{k=1}^\infty G_{n_k}$.

If $\mathcal{A}, \mathcal{B}$, and $a$ are as in Exercise 3 (with $\mathcal{A}$ not abelian and $\mathcal{B}$ a maximal abelian subalgebra of $\mathcal{A}$), show that $\sigma_{\mathcal{A}}(a) = \sigma_{\mathcal{B}}(a)$.

If $K$ is a nonempty compact subset of $\mathbb {C}$ that is polynomially convex, show that the components of $\text{int}\, K$ are simply connected.

Prove the following proposition.

Show that for $\mathcal{X}$ a Banach space and $A$ in $\mathcal{B}(\mathcal{X})$, $\sigma_l(A) = \sigma_r(A^*)$, where $\sigma_l$ and $\sigma_r$ denote the left and right spectra of $A$. What happens in a Hilbert space?

If $\mathcal{H}$ is an infinite dimensional Hilbert space and $K$ is a non-empty compact subset of $\mathbb {C}$, show that there is an $A$ in $\mathcal{B}(\mathcal{H})$ such that $\sigma (A) = K$. Can $A$ be found such that $\sigma (A) = \sigma_{ap}(A) = K$?

Let $K$ be a compact subset of $\mathbb {C}$. Does there exist an operator $A$ in $\mathcal{B}(C[0,1])$ such that $\sigma (A) = K$?

If $\mathcal{X}$ is a Banach space and $A \in \mathcal{B}(\mathcal{X})$, show that $A$ is left invertible if and only if $\ker A = (0)$ and $\text{ran}\, A$ is a closed complemented subspace of $\mathcal{X}$.

If $\mathcal{X}$ is a Banach space and $A \in \mathcal{B}(\mathcal{X})$, show that $A$ is right invertible if and only if $\text{ran}\, A = \mathcal{X}$ and $\ker A$ is a complemented subspace of $\mathcal{X}$.

If $\mathcal{X}$ is a Banach space and $T: \mathcal{X} \to \mathcal{X}$ is an isometry, then either $\sigma (T) \subseteq \partial \mathbb {D}$ or $\sigma (T) = \text{cl}\, \mathbb {D}$.

Verify the statements made in the following example. Let $\{ \alpha_n\} \in \ell^\infty$, $1 \le p \le \infty$, and define $A: \ell^p \to \ell^p$ by $(Ax)(n) = \alpha_n x(n)$. Then $\sigma (A) = \text{cl}\{ \alpha_n\}$ and $\sigma_p(A) = \{ \alpha_n\}$. For each $k$, define $N_k = \left\{ n \in \mathbb {N} : \alpha_n = \alpha_k\right\}$ and define $P_k: \ell^p \to \ell^p$ by $P_k x = \chi_{N_k} x$. If $\alpha_k$ is an isolated point of $\sigma (A)$, then $\{ \alpha_k\}$ is a clopen subset of $\sigma (A)$ and $E(\{ \alpha_k\} ; A) = P_k$.

Let $1 \le p \le \infty$ and suppose $0 < \alpha_1 \le \alpha_2 \le \cdots$ such that $r = \lim \alpha_n < \infty$. Define $A: \ell^p \to \ell^p$ by $A(x_1, x_2, \ldots ) = (0, \alpha_1 x_1, \alpha_2 x_2, \ldots )$. Show that $\sigma (A) = \left\{ z \in \mathbb {C} : \left|z\right| \le r\right\}$ and $\sigma_{ap}(A) = \partial \sigma (A)$. If $\left|\lambda \right| < r$, then $\text{ran}(A - \lambda )$ is closed and has codimension $1$. Also, $\sigma_p(A) = \square$.

Verify the statements made in the following example. A measurable function $k: [0,1] \times [0,1] \to \mathbb {C}$ is called a Volterra kernel if $k$ is bounded and $k(x, y) = 0$ when $x < y$. If $1 \le p \le \infty$ and $k$ is a Volterra kernel, define $V_k: L^p(0,1) \to L^p(0,1)$ by

Then $V_k \in \mathcal{B}(L^p)$ and $\left\| V_k\right\| \le \left\| k\right\|_\infty$. If $k, h$ are Volterra kernels and $(hk)(x, y) = \int_0^1 h(x, t) k(t, y) \, dt$, then $hk$ is a Volterra kernel, $\left\| hk\right\|_\infty \le \left\| h\right\|_\infty \left\| k\right\|_\infty$, and $V_{hk} = V_h V_k$. If $k$ is the characteristic function of $\left\{ (x,y) \in [0,1] \times [0,1] : y < x\right\}$, then $V_k$ is the Volterra operator. If $k$ is any Volterra kernel, then $\sigma (V_k) = \{ 0\}$.

Let $1 \le p \le \infty$ and let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space. For $\phi$ in $L^\infty (\mu )$, define $M_\phi$ on $L^p(\mu )$ to be the multiplication operator $M_\phi f = \phi f$. Find $\sigma (M_\phi )$, $\sigma_{ap}(M_\phi )$, and $\sigma_p(M_\phi )$.

If $A \in \mathcal{B}(\mathcal{X})$, $f \in \text{Hol}(A)$, and $\lambda \in \sigma_p(A)$, is $f(\lambda ) \in \sigma_p(f(A))$? If $\lambda \in \sigma_{ap}(A)$, is $f(\lambda ) \in \sigma_{ap}(f(A))$? Is there a relation between $f(\sigma_{ap}(A))$ and $\sigma_{ap}(f(A))$?

If $A \in \mathcal{B}(\mathcal{X})$, say that a complex number $\lambda$ has finite index if there is a positive integer $k$ such that $\ker (A - \lambda )^k = \ker (A - \lambda )^{k+1}$; the index of $\lambda$, denoted by $\nu (\lambda )$ or $\nu_A(\lambda )$, is the smallest such integer $k$.

If $V$ is the Volterra operator, show that $0$ is an essential singularity of $(z - V)^{-1}$.

Let $\mathcal{A}$ be a Banach algebra with identity. If $a \in \mathcal{A}$, define $L_a, R_a \in \mathcal{B}(\mathcal{A})$ by $L_a(x) = ax$ and $R_a(x) = xa$. Show that $\sigma (L_a) = \sigma (R_a) = \sigma (a)$.

If $E$ is a projection on a Hilbert space and $E$ is neither $0$ nor $1$, then $\sigma (E) = \{ 0, 1\}$.

(McCabe) If $\mathcal{X}$ is a complex Banach space and $T \in \mathcal{B}(\mathcal{X})$, show that the following statements are equivalent.

If $\mathcal{M}, \mathcal{N}$ are finite dimensional spaces and $\mathcal{M} \le \mathcal{N}$, $\mathcal{M} \ne \mathcal{N}$, then there is a $y$ in $\mathcal{N}$ such that $\left\| y\right\| = 1$ and $\text{dist}(y, \mathcal{M}) = 1$.

Let $A \in \mathcal{B}(\mathcal{X})$ and let $\lambda_1, \ldots , \lambda_n$ be distinct points in $\sigma_p(A)$. If $x_k \in \ker (A - \lambda_k)$, $1 \le k \le n$, and $x_k \ne 0$, show that $\{ x_1, \ldots , x_n\}$ is a linearly independent set.

Let $\mathcal{X}_1, \mathcal{X}_2, \ldots$ be Banach spaces and put $\mathcal{X} = \bigoplus_p \mathcal{X}_n$. Let $A_n \in \mathcal{B}(\mathcal{X}_n)$ such that $\sup_n \left\| A_n\right\| < \infty$ and define $A: \mathcal{X} \to \mathcal{X}$ by $A\{ x_n\} = \{ A_n x_n\}$. Show that $A \in \mathcal{B}(\mathcal{X})$ and $\left\| A\right\| = \sup_n \left\| A_n\right\|$. Show that $A \in \mathcal{B}_0(\mathcal{X})$ if and only if each $A_n \in \mathcal{B}_0(\mathcal{X})$ and $\lim \left\| A_n\right\| = 0$.