Fredholm theory

Prove that similar operators in Banach spaces are either both Fredholm with equal indices, or both not Fredholm.

Prove that every invertible operator in a Banach space is Fredholm.

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{K}(X, Y)$. Prove that $A \in \mathcal{F}(X, Y)$ if and only if both spaces $X$ and $Y$ are finite-dimensional.

Let $X$ be a Banach space and let $A$ be a Fredholm operator in $X$. Prove that then the operator $A^{\prime }$ is also Fredholm and $\alpha \left(A^{\prime }\right) = \beta (A), \beta \left(A^{\prime }\right) = \alpha (A)$.

Prove Theorems 11.5-11.8 for the case of a Hilbert space $H$ and a finite-dimensional operator $A \in \mathcal{B}(H)$.

Notation: for an operator $A \in \mathcal{K}(X)$ on a Banach space $X$ and the adjoint operator $A^{\prime } \in \mathcal{K}\left(X^{*}\right)$ one considers four equations: (1) $(I-A) x = y$; (2) $\left(I-A^{\prime }\right) f = g$; (3) $(I-A) x = 0$; (4) $\left(I-A^{\prime }\right) f = 0$. Let $M = \operatorname {Ker}(I-A)$ (the solutions of equation (3)) and $N = \operatorname {Ker}\left(I-A^{\prime }\right)$ (the solutions of equation (4)). For a pair $x \in X$, $f \in X^{*}$ one writes $x \perp f$ if $f(x) = 0$.

Theorem 11.5 (F. Riesz, 1918): If $A \in \mathcal{K}(X)$, then the operator $I-A$ is Fredholm.

Theorem 11.6 (first Fredholm theorem): Equation (1) (respectively, equation (2)) has a solution if and only if $y \perp N$ (respectively, $f \perp M$).

Theorem 11.7 (second Fredholm theorem, or the Fredholm alternative): Either equation (1) (respectively, equation (2)) has a solution for every right-hand side, or equation (3) (respectively, equation (4)) has a nonzero solution.

Theorem 11.8 (third Fredholm theorem): $\dim M = \dim N < \infty$.

Prove the first Fredholm theorem 11.6 for the case of a Hilbert space.

Theorem 11.6 (first Fredholm theorem): Let $A \in \mathcal{K}(X)$, $A^{\prime }$ the adjoint operator, $M = \operatorname {Ker}(I-A)$, $N = \operatorname {Ker}\left(I-A^{\prime }\right)$. The equation $(I-A) x = y$ (respectively, the equation $\left(I-A^{\prime }\right) f = g$) has a solution if and only if $y \perp N$ (respectively, $f \perp M$). Here for a pair $x \in X$, $f \in X^{*}$ one writes $x \perp f$ if $f(x) = 0$.

The statements of problems 11.71-11.75 contain theorems from the beginning of Section 11.4; this problem must be solved without using Theorems 11.3-11.8 (their statements are given in problems 11.69, 11.74 and 11.79).

Let $X$ be a Banach space, $A \in \mathcal{K}(X)$ and $T = I-A$. Prove that $\alpha (T) < \infty$.

The statements of problems 11.71-11.75 contain theorems from the beginning of Section 11.4; this problem must be solved without using Theorems 11.3-11.8 (their statements are given in problems 11.69, 11.74 and 11.79).

Let $X$ be a Banach space, $A \in \mathcal{K}(X)$ and $T = I-A$. Prove that the subspace $\operatorname {Im} T$ is closed in $X$.

The statements of problems 11.71-11.75 contain theorems from the beginning of Section 11.4; this problem must be solved without using Theorems 11.3-11.8 (their statements are given in problems 11.69, 11.74 and 11.79).

Let $X$ be a Banach space, $A \in \mathcal{K}(X)$ and $T = I-A$. Prove that $\beta (T) < \infty$.

The statements of problems 11.71-11.75 contain theorems from the beginning of Section 11.4; this problem must be solved without using Theorems 11.3-11.8 (their statements are given in problems 11.69, 11.74 and 11.79).

Prove Theorem 11.3.

Theorem 11.3: If $X$ and $Y$ are Banach spaces and $A \in \mathcal{F}(X, Y)$ (that is, $A \in \mathcal{B}(X, Y)$ is a Fredholm operator: the numbers $\alpha (A) = \dim \operatorname {Ker} A$ and $\beta (A) = \operatorname {codim} \operatorname {Im} A$ are finite), then $\operatorname {Im} A$ is closed (see problem 9.15).

The statements of problems 11.71-11.75 contain theorems from the beginning of Section 11.4; this problem must be solved without using Theorems 11.3-11.8 (their statements are given in problems 11.69, 11.74 and 11.79).

Let $X$ be a Banach space, $A \in \mathcal{K}(X)$ and $T = I-A$. Prove that:

Let $X, Y$ and $Z$ be Banach spaces, $A \in \mathcal{F}(Y, Z)$, $B \in \mathcal{F}(X, Y)$. Prove that $A B \in \mathcal{F}(X, Z)$ and ind $(A B) =$ ind $A+$ ind $B$.

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{I B}(X, Y)$ an invertible operator, $T \in \mathcal{K}(X, Y)$. Prove that $A+T \in \mathcal{F}(X, Y)$ and

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{F}(X, Y)$ and ind $A = 0$. Without using Theorem 11.4, prove F. Riesz's decomposition theorem: there exists an invertible operator $R \in \mathcal{I B}(X, Y)$ and a finite-dimensional operator $S$ such that $A = R+S$.

Theorem 11.4 (S. M. Nikol'skii, 1943): Let $X, Y$ be Banach spaces. An operator $A \in \mathcal{B}(X, Y)$ is Fredholm if and only if it is almost invertible, that is, there exist operators $R, L \in \mathcal{B}(Y, X)$ such that $L A = I_{X}+K_{1}$, $A R = I_{Y}+K_{2}$, where $K_{1} \in \mathcal{K}(X)$, $K_{2} \in \mathcal{K}(Y)$. Moreover, the operators $R$ and $L$ can be chosen equal and such that $K_{1}$ and $K_{2}$ are finite-rank (finite-dimensional) operators.

Prove Theorem 11.4.

Definition 11.5: Let $X, Y$ be Banach spaces. An operator $A \in \mathcal{B}(X, Y)$ is called almost invertible if there exist operators $R \in \mathcal{B}(Y, X)$, $L \in \mathcal{B}(Y, X)$ such that $L A = I_{X} + K_{1}$, $A R = I_{Y} + K_{2}$, where $K_{1} \in \mathcal{K}(X)$, $K_{2} \in \mathcal{K}(Y)$.

Theorem 11.4 (S. M. Nikol'skii, 1943): Let $X, Y$ be Banach spaces. An operator $A \in \mathcal{B}(X, Y)$ is Fredholm (that is, $\dim \operatorname {Ker} A < \infty$ and $\operatorname {codim} \operatorname {Im} A < \infty$) if and only if it is almost invertible. Moreover, the operators $R$ and $L$ can be chosen equal and such that $K_{1}$ and $K_{2}$ are finite-rank (finite-dimensional) operators.

(First index stability theorem). Let $X$ and $Y$ be Banach spaces, $S \in \mathcal{F}(X, Y), T \in \mathcal{K}(X, Y)$. Prove that $S+T \in \mathcal{F}(X, Y)$ and

(Second index stability theorem). Let $X$ and $Y$ be Banach spaces, $S \in \mathcal{F}(X, Y)$. Prove that there exists $\varepsilon > 0$ such that for every operator $T \in \mathcal{B}(X, Y)$ with $\left\| T\right\| < \varepsilon$ the inclusion $S+T \in \mathcal{F}(X, Y)$ holds together with the equality $\operatorname {ind}(S+T) = \operatorname {ind} S$.

Let $X \subset Y$ be Banach spaces. Prove that the embedding operator $J: X \rightarrow Y$ is Fredholm if and only if the codimension of the linear space $X$ in $Y$ is finite.

Prove that a projection operator in a Banach space is Fredholm if and only if its image has finite codimension.

Prove that the following operators are Fredholm:

Under what conditions on the sequence $\left\{ \lambda_{n}\right\}_{1}^{\infty } \in l_{\infty }$ is the operator $A x = \left(\lambda_{1} x_{1}, \lambda_{2} x_{2}, \ldots , \lambda_{n} x_{n}, \ldots \right)$ in $l_{p}$, where $1 \leqslant p < \infty$, a Fredholm operator? In those cases where the operator is Fredholm, compute $\alpha (A), \beta (A)$ and ind $A$.

Under what conditions on the function $a$ is the operator $(A x)(t) = a(t) x(t)$ Fredholm:

Prove that the operators of right and left shift in any of the spaces $l_{p}, p \in [1, \infty ]$, $c$ or $c_{0}$ are Fredholm. Compute the deficiency numbers and index for these operators.

Prove that the operator $(A x)(t) = x(t+a)$ is Fredholm in any of the spaces $L_{p}(\mathbb {R}), p \in [1, \infty ], B C(\mathbb {R})$. Compute $\alpha (A)$, $\beta (A)$ and $\operatorname {ind} A$ for this operator in each of the indicated spaces.

Let $\left\{ e_{n}\right\}_{1}^{\infty }$ be an orthonormal basis in a Hilbert space $H$, and let the system $\left\{ x_{n}\right\}_{1}^{\infty }$ be quadratically close to it, i.e., $\sum_{n = 1}^{\infty }\left\| x_{n}-e_{n}\right\|^{2} < \infty$. Prove that under these conditions the following statements are equivalent:

1. the system $\left\{ x_{n}\right\}_{1}^{\infty }$ is complete;

2. the system $\left\{ x_{n}\right\}_{1}^{\infty }$ is minimal;

3. the system $\left\{ x_{n}\right\}_{1}^{\infty }$ is a Riesz basis. ¹