Fredholm Alternative

Let \(a\) and \(b\) be non-zero (but possibly complex) elements in the complex Hilbert space \(\ell ^2\). Consider the bounded linear operator \(T : \ell ^2 \to \ell ^2\) defined by

1. Use the Fredholm Alternative Theorem to find a simple condition on \(a\) and \(b\) guar- anteeing that the equation

   \[(T - I)u = v\]

   has a unique solution \(u\) for all \(v \in \ell ^2\).

2. Find the Hilbert adjoint operator \(T^* : \ell ^2 \to \ell ^2\).

3. Given \(T : X \to Y\) a bounded linear operator, recall that the adjoint operator \(T^\times : Y'\to X'\) is defined

   \[(T^\times g)(x) = g(Tx).\]

   Find the adjoint operator \(T^\times : \ell ^2 \to \ell ^2\). Hint: Express \(g\) and \(T^\times g\) using the Riesz Representation Theorem.

4. Use the Fredholm Alternative Theorem to give conditions on \(v_0\) guaranteeing that

   \[(T - I)u = v_0\]

   has a unique solution \(u\). When it exists, find the solution.

Let \(X\) be a Banach space and \(T : X \to X\) be a compact linear operator. Use the Fredholm alternative to show that if \(\lambda \notin \sigma _p(T)\) and \(\lambda \neq 0\) then \(\lambda \in \rho (T)\), or in other words: that all non-zero spectral values of a compact operator are eigenvalues.

Let \(a\in \ell ^2\setminus \{0\}\). Consider the bounded linear operator \(T:\ell ^2\to \ell ^2\) defined by \[ Tu=\langle u,a\rangle a. \]

1. Show that the operator \(T\) is compact and self-adjoint.
2. Use the Fredholm Alternative Theorem to find a simple condition on \(a\) guaranteeing that the equation \[ (T-I)u=v \] has a unique solution \(u\) for all \(v\in \ell ^2\). What is the solution?

Let \(X\) be a Banach space, let \(z\in X\) with \(z\ne 0\), and let \(f\in X'\) with \(f\ne 0\). Consider the linear operator \(T:X\to X\) defined for \(x\in X\) by \[ Tx=f(x)z. \]

1. Briefly explain why \(T\) is compact.
2. Use Neumann series to give a condition on \(f\) and \(z\) that ensures the equation \(x+Tx=y\) admits a unique solution \(x\) for all \(y\in X\).
3. Use the Fredholm alternative to find a condition on \(f\) and \(z\) that ensures the equation \(x+Tx=y\) admits a unique solution \(x\) for all \(y\in X\).

Define \[ Ku(x)=\sum _{n=0}^{10}\left (\int _{-1}^{1}e^{-ny}u(y)\,dy\right )x^n, \qquad x\in [-1,1]. \]

1. Find the adjoint \(K^*\) of the operator \(K\).
2. Assuming \(\lambda \ne 0\) is fixed and real, use the Fredholm Alternative to characterize the set of functions \(g\) such that the problem \((K-\lambda I)u=g\) on \([-1,1]\) has a solution.
3. From the structure of the operator \(K\), what can be said about the set \[ \sigma _p=\{\lambda \in \mathbb {C}:(K-\lambda I)u=0\text { has a solution }u\ne 0\}? \] There is no need to actually calculate \(\sigma _p\).

Consider the problem \(Lu=1\) on \((0,\pi )\), where \(Lu=u''+u\) with boundary conditions \(u'(0)=u'(\pi )=0\).

1. Write down the weak formulation of the problem.
2. Find the \(L^2[0,\pi ]\) adjoint of \(L\), and determine whether or not \(L\) is self-adjoint.
3. Find the nullspace of \(L\). Use Fredholm theory to characterize the set of functions \(f\) such that \(Lu=f\) has a solution. Verify that \(f=1\) satisfies your condition, and find an infinite set of solutions to the problem \(Lu=1\).

1. State and prove the Fredholm Alternative Theorem for a linear transformation \(A:\mathbb {R}^n\to \mathbb {R}^m\).
2. Find and give a graphical interpretation of the pseudo-inverse of the matrix \[ A= \begin {pmatrix} 1 & 1\\ 2 & 2 \end {pmatrix}. \]

Consider the integral equation \[ \phi (x)-\lambda \int _0^\pi \sin (x+t)\phi (t)\,dt = \cos (x)+\sin (x), \qquad 0\le x\le \pi . \]

1. Find the unique solution when \(\lambda \ne \pm 2/\pi \).
2. Use the Fredholm alternative theorem to show that there is no solution when \(\lambda =2/\pi \).
3. Use the Fredholm alternative theorem to show that when \(\lambda =-2/\pi \) there is a one-parameter family of solutions of the form \[ \phi (x)=\frac {\cos (x)+\sin (x)}{2}+\alpha (\cos (x)-\sin (x)). \]

Suppose that \(L\) is a bounded linear operator in a Hilbert space \(H\) with closed range.

1. Prove the Fredholm alternative theorem for solutions of the inhomogeneous equation \(Lu=f\) with \(u,f\in H\).
2. Suppose that \(L\) is invertible and has a complete orthonormal set of eigenfunctions \(\phi _n\), integer \(n\). Solve the inhomogeneous equation of part (a) using an eigenfunction expansion.
3. Suppose that the adjoint operator \(L^*\) has a nontrivial nullspace \(\mathcal {N}(L^*)\) spanned by the functions \(\psi _i\), \(i=1,\ldots ,m\) and \(f\) does not lie in the orthogonal complement of \(\mathcal {N}(L^*)\). Let \(L\) have a nullspace spanned by the functions \(\phi _i\), \(i=1,\ldots ,m\). Explain the construction of the smallest least squares solution.

Find the eigenvalues and eigenfunctions of the integral operator \[ (Ku)(x)=\int _0^1 k(x,y)u(y)\,dy \] where \[ k(x,y)=\min \{x,y\}. \] Solve the corresponding integral equation \[ u(x)-\lambda \int _0^1 k(x,y)u(y)\,dy=f(x) \] using the eigenfunctions of \(K\) assuming that \(\lambda \ne 1/a_n^2\) for any integer \(n\), where \(a_n=(2n+1)\pi /2\) (i.e. determine the resolvent operator). Write down the Fredholm alternative condition for \(f\) when \(\lambda =1/a_n^2\).

Find the eigenvalues and eigenfunctions of the integral operator \[ (Ku)(x)=\int _0^1 k(x,y)u(y)\,dy \] where \[ k(x,y)= \begin {cases} x(1-y), & 0\le x<y\le 1,\\ y(1-x), & 0\le y<x\le 1. \end {cases} \] Solve the corresponding integral equation \[ u(x)-\lambda \int _0^1 k(x,y)u(y)\,dy=f(x) \] using the eigenfunctions of \(K\) assuming that \(\lambda \ne n^2\pi ^2\) for any integer \(n\) (i.e. determine the resolvent operator). Write down the Fredholm alternative condition for \(f\) when \(\lambda =m^2\pi ^2\) for some integer \(m\).

State and prove the Fredholm Alternative Theorem for linear operators \(L:\mathbb {R}^n\to \mathbb {R}^m\).

Let \(A\) be a real \(m\times n\) matrix with adjoint \(A^*\).

1. Prove that if \(R(A)\) and \(N(A)\) denote the range and nullspace of \(A\) respectively, then \[ R(A)\oplus N(A^*)=\mathbb {R}^m. \] (\(\oplus \) means orthogonal direct sum.)
2. Using the result of (a), state and prove Fredholm’s Alternative Theorem.