The Spectrum

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 $\sigma (ab) \ne \sigma (ba)$.