The Spectrum of a Linear Operator

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.