Compact operators and Fredholm theory

Prove that an operator $A \in \mathcal{B}(X, Y)$ is compact if and only if the image of the unit ball in $X$ is a precompact set in $Y$.

Prove that a finite linear combination of compact operators is a compact operator.

Prove that the composition of a compact and a bounded operator (in either order) is a compact operator.

Prove that if $\left\{ A_{n}\right\}_{1}^{\infty }$ are compact operators and $A_{n} \rightrightarrows A$, then $A$ is also a compact operator.

(J. Calkin, 1941). Let H be a separable Hilbert space. Prove that the unique nontrivial proper closed two-sided ideal in $\mathcal{B}(H)$ is $\mathcal{K}(H)$.

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{K}(X, Y)$. Prove that $A$ can have a right inverse operator only in the case $\operatorname {dim} Y < \infty$. Prove that $A$ can have a left inverse operator only in the case $\operatorname {dim} X < \infty$.

Give an example of a sequence of compact operators $\left\{ A_{n}\right\}_{1}^{\infty }$ in a Banach space $X$ such that $A_{n} \xrightarrow {s} A \in \mathcal{B}(X)$, but $A$ is not a compact operator.

Let $X$ and $Y$ be normed spaces, $A \in \mathcal{B}(X, Y)$. Prove that if $\operatorname {dim}(\operatorname {Im} A) < \infty$, then $A$ is compact.

Let $X$ be a Banach space, $A \in \mathcal{K}(X, Y)$, and let $Y_{0} \subset \operatorname {Im} A$ be a closed subspace. Prove that $\operatorname {dim} Y_{0} < \infty$.

Give an example of Banach spaces $X$ and $Y$ and a non-compact operator $A \in \mathcal{B}(X, Y)$ such that $\operatorname {Im} A$ contains no infinite-dimensional closed subspaces.

Let $A \in \mathcal{B}(X)$, where $X$ is a Banach space, and $\operatorname {Im} A = X$. Can the operator $A$ be compact?

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{K}(X, Y), B \in \mathcal{B}(X, Y)$ and $\operatorname {Im} B \subset \operatorname {Im} A$. Prove that $B \in \mathcal{K}(X, Y)$.

Let $X$ and $Y$ be normed spaces, the operator $A \in \mathcal{K}(X, Y)$. Prove that the space $(\operatorname {Im} A,\left\| \cdot \right\|_{Y})$ is separable.

Let $X$ be a Banach space. Prove that the identity operator on it is compact if and only if $X$ is finite-dimensional.

Prove that a projection operator in a Banach space is compact if and only if it is finite-dimensional.

Let $X$ be a reflexive Banach space, let $X^{*}$ be separable, and $A \in \mathcal{K}(X)$. Prove that the image of the closed unit ball $A(\bar{B}(0,1))$ is a compact set. ¹

Theorem 14.4 (S. Banach, 1932; L. Alaoglu, 1940): Let $X$ be a linear normed space. The closed unit ball of $X$ is compact in the weak topology if and only if $X$ is reflexive.

Let $X$ be a Banach space. Prove that if for every operator $A \in \mathcal{K}(X)$ the image of the closed unit ball $A(\bar{B}(0,1))$ is a compact set, then the space $X$ is reflexive.

Is it true that if in an infinite-dimensional normed space $A^{2} = 0$, then the operator $A$ is compact?

Let $X$ be an infinite-dimensional Banach space, the operator $A \in \mathcal{K}(X)$, and the polynomial $P(z) = a_{n} z^{n}+a_{n-1} z^{n-1}+\ldots +a_{0}$. Prove that if $a_{0} \neq 0$, then $P(A) \neq 0$. For an arbitrary polynomial $P(z)$ with coefficient $a_{0} = 0$, give an example of a compact operator $A$ for which $P(A) = 0$.

Prove that similar operators (see Definition 9.4) are either both compact or both non-compact.

Definition 9.4: Let $X$ and $Y$ be normed spaces, $A \in \mathcal{B}(X)$, $B \in \mathcal{B}(Y)$. The operators $A$ and $B$ are called similar if there exists a bounded operator $S$ with bounded inverse (that is, $S \in \mathcal{I B}(Y, X)$) such that $A S = S B$ (notation $A \sim B$).

Prove that a compact operator carries every weakly convergent sequence into a sequence convergent in norm.

Prove that an operator in a reflexive space carrying every weakly convergent sequence into a sequence convergent in norm is compact. Give an example of an operator in a nonreflexive space carrying every weakly convergent sequence into a sequence convergent in norm, but which is not compact.

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{K}(X, Y)$, the sequence $x_{n} \rightharpoonup x$ in $X$, the sequence $f_{n} \stackrel{*}{\rightharpoonup } f$ in $Y^{*}$. Prove that $f_{n}\left(A x_{n}\right) \rightarrow f(A x)$.

Let $X$ and $Y$ be Banach spaces, with $Y$ separable. Let $A \in \mathcal{B}(X, Y)$, and let the operator $A^{\prime }$ carry every $*$-weakly convergent sequence in $Y^{*}$ into a sequence convergent in norm in $X^{*}$. Prove that $A$ is compact.

Let a Banach space $X$ be reflexive, and let $X^{*}$ be separable. Prove that every compact operator $A \in \mathcal{K}(X)$ attains its norm on the closed unit ball ¹. Give an example of a compact operator in a nonreflexive space that does not attain its norm on the closed unit ball.

Theorem 14.4 (S. Banach, 1932; L. Alaoglu, 1940): Let $X$ be a linear normed space. The closed unit ball of $X$ is compact in the weak topology if and only if $X$ is reflexive.

Prove that $\operatorname {Sp}(X, Y)$ is a (in general, non-closed) linear subspace of $\mathcal{K}(X, Y)$. Prove that $\operatorname {dim} \operatorname {Sp}(X, Y) = \infty$, if $\operatorname {dim} X = \infty$ or $\operatorname {dim} Y = \infty$.

On the linear space $\operatorname {Sp}(X, Y)$ introduce the norm

where the infimum is taken over all representations of the operator $A$ as the sum of an absolutely convergent series $A = \sum_{k = 1}^{\infty } A_{k}$ of one-dimensional operators. Prove that this function satisfies the axioms of a norm. Prove the estimate $\left\| A\right\|_{\mathcal{B}(X, Y)} \leqslant \left\| A\right\|_{\mathrm{Sp}(X, Y)}$. Prove that the space $\operatorname {Sp}(X, Y)$ is Banach with respect to the introduced norm.

Prove that any Banach space in which there is a countable Schauder basis has the approximation property. Prove that any (not necessarily separable) Hilbert space has the approximation property.

Suppose that a Schauder basis exists in a Banach space $X$. Prove that the closure of the subspace of finite-dimensional operators in the sense of strong operator convergence in $\mathcal{B}(X)$ coincides with $\mathcal{B}(X)$.

For which $\lambda \in l_{\infty }$ is the operator $A x = \left(\lambda_{1} x_{1}, \ldots , \lambda_{n} x_{n}, \ldots \right)$ a compact operator in $l_{p}(1 \leqslant p < \infty )$? Under what conditions on the sequence $\lambda$ is this operator nuclear?

For which $a \in L_{\infty }[a, b]$ is the operator $(A x)(t) = a(t) x(t)$ a compact operator in $L_{p}[0,1], p \in [1, \infty ]$? For which $a \in C[0,1]$ is the operator $(A x)(t) = a(t) x(t)$ compact in $C[0,1]$?

Prove that any bounded operator acting from a reflexive space into the space $l_{1}$ is compact. In particular, if $A \in \mathcal{B}\left(l_{p}, l_{1}\right), 1 < p < \infty$, then $A \in \mathcal{K}\left(l_{p}, l_{1}\right)$.

Prove that any bounded operator acting from the space $c_{0}$ into a reflexive space is compact.

Let $A \in \mathcal{B}\left(L_{2}[0,1]\right)$ and $\operatorname {Im} A \subset C[0,1]$. Prove that $A \in \mathcal{K}\left(L_{2}[0,1]\right)$ (cf. problem 9.18).

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 not compact.

Prove that if $K \in C\left([a, b]^{2}\right)$, then the integral operator $A: L_{p}[a, b] \rightarrow L_{r}[a, b],(A x)(t) = \int_{a}^{b} K(t, s) x(s) d s$, is compact for any $p, r \in [1, \infty )$.

Prove that if $K \in L_{q}\left([a, b]^{2}\right)$, then the integral operator $A: L_{p}[a, b] \rightarrow L_{q}[a, b], \frac{1}{p}+\frac{1}{q} = 1, q < \infty ,(A x)(t) = \int_{a}^{b} K(t, s) x(s) d s$, is compact.

Prove that if the function $K(t, s)$ is continuous on the square $[a, b]^{2}$ except for a finite number of continuous curves of the form $s = s_{k}(t), k = 1, \ldots , n, t \in [a, b]$, then the integral operator $A: C[a, b] \rightarrow C[a, b],(A x)(t) = \int_{a}^{b} K(t, s) x(s) d s$, is compact.

Prove that the differentiation operator $A x = x^{\prime }$ is not compact when acting:

Prove that the shift operator $(A x)(t) = x(t-a)$ is not compact in the spaces:

Using problem 11.21, prove that the Hardy operator

in the space $L_{p}[0,1], p \in (1, \infty )$, is not compact.

Give an example of a non-compact bounded operator $A$ in the space $L_{p}[0,1], p \in (1, \infty )$, and a sequence $\left\{ x_{n}\right\}_{1}^{\infty } \subset L_{p}[0,1]$ with the properties

1. $x_{n} \rightharpoonup 0$;

2. $\left\| x_{n}\right\|_{L_{p}} = 1$ for all $n \in \mathbb {N}$;

3. $\left\| A x_{n}\right\|_{L_{p}} \rightarrow 0$ (cf. problem 11.22).

Let $X$ be one of the spaces: $l_{p}, p \in (1, \infty )$, $c$ or $c_{0}$. Let $A \in \mathcal{K}(X)$, and $\left\{ e_{n} = (0,0, \ldots , 0,\underset {n}{1},0,0, \ldots )\right\}_{n = 1}^{\infty }$. Prove that $A e_{n} \rightarrow 0$. Give an example of a compact operator $A$ in the space $l_{1}$ such that $A e_{n} \nrightarrow 0$.

Prove that the operator $A$ of the form

in the spaces $l_{p}, p \in (1, \infty )$, $c$, and $c_0$, is bounded, not compact, but $\left\| A e_{n}\right\| \rightarrow 0$.

Is the operator $(A x)(t) = \frac{1}{2}(x(t)+x(-t))$ compact in the space $C[-1,1]$?

Let $\varphi (t)$ be a continuous map of the segment $[0,1]$ into itself. Prove that the composition operator $A \in \mathcal{B}(C[0,1]),(A x)(t) = x(\varphi (t))$, is compact in $C[0,1]$ if and only if $\varphi (t) \equiv$ const.

(Theorem of I. K. Daugavet, 1963). Prove that for every compact operator $A$ in $C[0,1]$ the equality $\left\| A+\lambda I\right\| = \left\| A\right\| +\left|\lambda \right|(\lambda \in \mathbb {C})$ holds.

Let, in a Hilbert space $H$, the operator $A^{*} A$ be compact. Prove that the operator $A$ is compact.

Prove that in a Hilbert space the operator $A$ is compact if and only if $A^{*}$ is compact.

Let $A$ be a self-adjoint nonnegative operator in a Hilbert space $H$. Prove that $A$ is compact if and only if $\sqrt{A}$ is compact.

Let $A, B$ and $C$ be self-adjoint operators in a Hilbert space, $B \leqslant A \leqslant C$, where $B$ and $C$ are compact. Prove that $A$ is also compact.

For which values of the parameter $r$ is the Hardy operator

(see problem 6.33), compact?

Prove that if $A \in \mathcal{K}(H)$, and $\left\{ e_{n}\right\}_{n = 1}^{\infty }$ is an orthonormal basis of the space $H$, then $\left\| A e_{n}\right\| \rightarrow 0$.

Prove the compactness of the operator $A: l_{2} \rightarrow l_{2}$,

Is the operator $A$ compact in $l_{2}$ under the condition $\lim_{i \rightarrow \infty } \sum_{j = 1}^{+\infty }\left|a_{i j}\right|^{2} = 0$?

Is the Hilbert operator (see problem 6.72), defined by the matrix $\left\{ a_{i j}\right\}_{i, j = 1}^{\infty }$, where $a_{i j} = \frac{1}{i+j}$, compact in the space $l_{2}$?

Give an example of a compact operator $A$ in the space $l_{2}$ whose matrix entries in the standard basis satisfy the relation $\sum_{k = 1}^{\infty } \sum_{n = 1}^{\infty }\left|a_{k n}\right|^{2} = \infty$.

Let $T_{r}:\left(x_{1}, x_{2}, \ldots \right) \mapsto \left(0, x_{1}, x_{2}, \ldots \right)$ be the right shift operator in the space $l_{2}$. Prove that if a compact operator $K$ commutes with $T_{r}$, then $K = 0$.

Prove that the operator $(A x)(t) = \int_{-\infty }^{+\infty } e^{-\left|t-s\right|} x(s) d s$ is not compact in the space $L_{2}(\mathbb {R})$.

Prove the compactness in the space $L_{2}[a, b]$ of the integral operator $(A x)(t) = \int_{a}^{b} K(t, s) x(s) d s$ with kernel $K \in L_{2}\left([a, b]^{2}\right)$.

For which $\alpha \in (0,1)$ is the operator

compact?

Prove the compactness of the operator $A: L_{2}[0,1] \rightarrow L_{2}[0,1]$, $(A x)(t) = \int_{0}^{1} \frac{1}{t-s} \ln \frac{1+t}{1+s} x(s) d s$.

Let $H$ be a Hilbert space, $A \in \mathcal{K}(H)$. Prove that there exists a vector $x \in H,\left\| x\right\| = 1$, such that $\left\| A x\right\| = \left\| A\right\|$ (cf. problem 11.25).

Let $H$ be a Hilbert space, $A \in \mathcal{K}(H)$, a vector $x \in H,\left\| x\right\| = 1$, and $\left\| A x\right\| = \left\| A\right\|$. Prove that the operator $A$ carries $\left\{ x\right\}^{\perp }$ into $\left\{ A x\right\}^{\perp }$.

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

Prove that the general solution of the equation

(with respect to the function $x$, with $y$ known) has the form

here $c_{i} \in \mathbb {C}, 1 \leqslant i \leqslant n$, are solutions of the system of equations

where the coefficients $a_{i j}$ and $b_{i}$ depend on $y, p_{i}$ and $q_{i}$. State the Fredholm alternative in terms of the matrix $a_{i j}$ and the right-hand side $b_{i}$. Assume the functions $p_{i}$ to be linearly independent.

In the space $C[0,1]$, find the solution of the following integral equations for all values of the complex parameters $\lambda , \alpha , \beta$ and $\gamma$:

In the space $C[0, \pi ]$, find the solution of the following integral equations for all values of the complex parameters $\lambda , \alpha$ and $\beta$: