Fredholm Theory

Let $S$ be the unilateral shift of multiplicity $1$ on $l^2(\mathbb {N})$ and find $\sigma_l(S)$ and $\sigma_r(S)$.

The compression spectrum of $A$, $\sigma_c(A)$, is defined by $\sigma_c(A) = \left\{ \lambda \in \mathbb {C}: \mathrm{ran}(A - \lambda ) \text{ is not dense in } \mathcal{H}\right\}$. Show:

If $A \in \mathcal{B}(\mathcal{H})$ and $f \in \mathrm{Hol}(A)$, then $f(\sigma_p(A)) \subseteq \sigma_p(f(A))$. If $f$ is not constant on any component of its domain, then $f(\sigma_p(A)) = \sigma_p(f(A))$.

If $A \in \mathcal{B}(\mathcal{H})$ and $f \in \mathrm{Hol}(A)$, then $f(\sigma_{ap}(A)) = \sigma_{ap}(f(A))$.

If $A \in \mathcal{B}(\mathcal{H})$ and $\mathrm{ran}\ A$ is closed, prove that $\mathrm{ran}\ A^*$ is closed without using Theorem VI.1.10. [Hint: Show that there is a bounded operator $B$ on $\mathcal{H}$ such that $BA =$ the projection of $\mathcal{H}$ onto $(\mathrm{ker}\ A)^\perp$.]

Theorem 2.3 states that if $A: \mathcal{H} \to \mathcal{H}'$ is a bounded operator, then the following two statements (among others) are equivalent: (a) $A$ is left semi-Fredholm, that is, there is a bounded operator $B: \mathcal{H}' \to \mathcal{H}$ and a compact operator $K$ on $\mathcal{H}$ such that $BA = 1 + K$; (b) $\mathrm{ran}\ A$ is closed and $\dim \mathrm{ker}\ A < \infty$. Give a direct proof that (b) implies (a).

Let $A, B, C \in \mathcal{B}(\mathcal{H})$ and define $X: \mathcal{H}^{(2)} \to \mathcal{H}^{(2)}$ by the matrix

If $A \in \mathcal{B}(\mathcal{H})$, show that $A\mathcal{M}$ is closed for every closed subspace $\mathcal{M}$ of $\mathcal{H}$ if and only if $A$ has finite rank or $A$ is left Fredholm.

If $\mathscr {X}$ and $\mathscr {Y}$ are Banach spaces and $T \in \mathcal{B}(\mathscr {X}, \mathscr {Y})$ such that the Hamel basis dimension (that is, the algebraic dimension) of $\mathscr {Y}/(\mathrm{ran}\ T)$ is finite, then $\mathrm{ran}\ T$ is closed.

Show that a normal operator is Fredholm if and only if $0$ is not a limit point of $\sigma (N)$ and $\dim \mathrm{ker}\ N < \infty$. (See Proposition 4.5 below.)

Prove Lemma 3.6: let $\mathcal{M}$ and $\mathcal{N}$ be two closed subspaces of the Hilbert space $\mathcal{H}$.

If $A \in \mathcal{B}(\mathcal{H})$ and $\mathrm{ran}\ A$ is closed, show that $\mathrm{ran}\ A^{(\infty )}$ is closed. If $A \in \mathcal{SF}$ and $\mathrm{ker}\ A = (0)$, show that $A^{(\infty )} \in \mathcal{SF}$ and $\operatorname {ind} A^{(\infty )} = -\infty$ or $0$.

Does the unilateral shift of multiplicity $1$ have a square root?

Show that for every $n$ in $\mathbb {Z} \cup \left\{ \pm \infty \right\}$ there is an operator $A$ in $\mathcal{SF}$ such that $\operatorname {ind} A = n$.

If $A \in \mathcal{SF}$, then for every $n \ge 1$, $A^n \in \mathcal{SF}$ and $\operatorname {ind} A^n = n(\operatorname {ind} A)$.

If $A: \mathcal{H} \to \mathcal{H}'$ is a left semi-Fredholm operator, then there is a finite rank operator $F: \mathcal{H} \to \mathcal{H}'$ such that $\mathrm{ker}(A + F) = (0)$ and $\operatorname {ind}(A + F) = \operatorname {ind} A$.

If $A$ is a Fredholm operator in $\mathcal{B}(\mathcal{H})$, prove that the following statements are equivalent.

1. $\operatorname {ind} A = 0$.

2. There is a compact operator $K$ such that $A + K$ is invertible.

3. There is a finite rank operator $F$ such that $A + F$ is invertible.

If $U$ is the unilateral shift of multiplicity $1$ and $\pi : \mathcal{B}(\mathcal{H}) \to \mathcal{B}(\mathcal{H})/\mathcal{B}_0(\mathcal{H})$ is the natural map, show that $\pi (U)$ is normal in $\mathcal{B}(\mathcal{H})/\mathcal{B}_0(\mathcal{H})$ but there is no normal operator $N$ such that $U - N$ is compact. (See Exercise IX.8.14.)

Show that the material of this section is only significant for infinite dimensional Hilbert spaces by showing that the essential spectrum of every operator on $\mathcal{H}$ is non-empty if and only if $\mathcal{H}$ is infinite dimensional.

Let $G$ be a bounded region in $\mathbb {C}$ such that $\partial G = \partial [\mathrm{cl}\ G]$ and let $\phi$ be a function that is analytic in a neighborhood of $\mathrm{cl}\ G$. Define $A: L_a^2(G) \to L_a^2(G)$ by $Af = \phi f$. Find all of the parts of the spectrum of $A$.

(Fillmore, Stampfli, and Williams [1972].) If $\lambda \in \sigma_{le}(A)$, then there is a projection $P$, having infinite rank, such that $\pi (A - \lambda )\pi (P) = 0$.

(Fillmore, Stampfli, and Williams [1972].) Let $A \in \mathcal{B}(\mathcal{H})$.

Let $A \in \mathcal{F}$ and suppose $f$ is analytic in a neighborhood of $\sigma (A)$ and does not vanish on $\sigma_e(A)$. Show that $f(A) \in \mathcal{F}$ and find $\operatorname {ind} f(A)$.

Let $S$ be the unilateral shift and let $f$ be an analytic function in a neighborhood of $\mathrm{cl}\ \mathbb {D}$ such that $f(z) \ne 0$ if $\left|z\right| = 1$. Let $\gamma (t) = f(\exp (2\pi i t))$, $0 \le t \le 1$. Show that $\sigma_e(f(S)) = f(\partial \mathbb {D}) = \left\{ \gamma (t): 0 \le t \le 1\right\}$ and that if $\lambda \notin f(\partial \mathbb {D})$, $\operatorname {ind}(f(S) - \lambda ) = -n(\gamma ; \lambda )$, where $n(\gamma ; \lambda ) =$ the winding number of $\gamma$ about $\lambda$. Moreover, show that if $\operatorname {ind}(f(S) - \lambda ) = 0$, then $\lambda \notin \sigma (f(S))$.

Let $S$ be the operator defined in Example 2.11 where $G = \mathbb {D}$. Show that there is a compact operator $K$ such that $S + K$ is unitarily equivalent to the unilateral shift.

If $S$ is the unilateral shift, show that for every $\varepsilon > 0$ there is a rank one operator $F$ with $\left\| F\right\| < \varepsilon$ such that $\sigma (S^* \oplus S + F) = \partial \mathbb {D}$.

Let $G$ be an open connected subset of $\sigma (A) \setminus \sigma_{le}(A) \cup \sigma_{re}(A)$ and suppose $\lambda_0 \in G$ such that $\operatorname {ind}(A - \lambda_0) = 0$. Show that there is a finite rank operator $F$ such that $A + F - \lambda_0$ is invertible. Show that $A + F - \lambda$ is invertible for every $\lambda$ in $G$.

Let $G$ be any topological group and let $G_0$ be the component of the identity. Show that $G_0$ is a normal subgroup of $G$.

What are the components of the set of invertible elements in $C(\partial \mathbb {D})$?

If $S =$ the unilateral shift, what are the components of the set of invertible elements of $C^*(S)$?

If $U$ and $V$ are isometries, show that there is a path consisting entirely of isometries connecting $U$ to $V$ if and only if $\dim (\mathrm{ran}\ U)^\perp = \dim (\mathrm{ran}\ V)^\perp$.

Find the components of the set of partial isometries.

If $A$ is an invertible operator, show that $\gamma (A) = \left\| A^{-1}\right\|^{-1}$.

Let $A \in \mathcal{F}$ and suppose that $\mathrm{ker}\ A \ne (0)$ and $\mathrm{ran}\ A^\perp \ne (0)$. Show that for every $\delta > 0$ there is an operator $B$ in $\mathcal{F}$ such that $\left\| B - A\right\| < \delta$, $\dim \mathrm{ker}\ B < \dim \mathrm{ker}\ A$, and $\dim \mathrm{ran}\ B^\perp < \dim \mathrm{ran}\ A^\perp$.

If $A \in \mathcal{SF}$, show that there is a $\delta > 0$ such that $\dim \mathrm{ker}(A - \mu ) = \dim \mathrm{ker}\ A$ and $\dim \mathrm{ran}(A - \mu )^\perp = \dim \mathrm{ran}\ A^\perp$ for $\left|\mu \right| < \delta$ if $\mathrm{ker}\ A \subseteq \mathrm{ran}\ A^n$ for every $n \ge 1$.

Prove Proposition 6.9: if $\lambda$ is an isolated point of $\sigma (A)$, the following statements are equivalent.

1. $\lambda \notin \sigma_{le}(A) \cap \sigma_{re}(A)$.

2. The Riesz idempotent $E(\lambda )$ has finite rank.

3. $A - \lambda \in \mathcal{F}$ and $\operatorname {ind}(A - \lambda ) = 0$.

Prove Proposition 6.10: if $A \in \mathcal{B}(\mathcal{H})$, then $\sigma_e(A) = [\sigma_{le}(A) \cap \sigma_{re}(A)] \cup P_{+\infty }(A) \cup P_{-\infty }(A)$, where, for $n \in \mathbb {Z} \cup \left\{ \pm \infty \right\}$, $P_n(A) \equiv \left\{ \lambda \in \sigma (A): A - \lambda \in \mathcal{SF} \text{ and } \operatorname {ind}(A - \lambda ) = n\right\}$.

If $\lambda \in \partial P_n(A)$ and $n \ne 0$, show that $\mathrm{ran}(A - \lambda )$ is not closed. What happens if $n = 0$?

(Stampfli [1974].) If $A \in \mathcal{B}(\mathcal{H})$, then there is a $K$ in $\mathcal{B}_0(\mathcal{H})$ such that $\sigma (A + K) = \sigma_w(A)$.

Prove Proposition 6.13: let $K$ be a compact subset of $\mathbb {C}$, let $\left\{ G_n: -\infty \le n \le \infty \right\}$ be disjoint open subsets of $K$ (some possibly empty), let $D$ be a subset of the set of isolated points of $K$, and for each $\lambda$ in $D$ let $n_\lambda \in \left\{ 1, 2, \ldots \right\}$. Then there is an operator $A$ on $\mathcal{H}$ such that $\sigma (A) = K$, $P_n(A) = G_n$ for $0 < \left|n\right| \le \infty$, $P_0(A) = G_0 \cup D$, and $\dim E(\lambda ) = n_\lambda$ for every $\lambda$ in $D$.

(Conway [1985].) If $L$ and $R$ are nonempty compact subsets of $\mathbb {C}$, then there is a bounded operator $A$ on $\mathcal{H}$ such that $\sigma_l(A) = L$ and $\sigma_r(A) = R$ if and only if $\partial L \subseteq R$ and $\partial R \subseteq L$.