Adjoint Operators and Riesz Representation

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 \(H\) be a Hilbert space and \(A : H \to H\) be a bounded self-adjoint linear operator.

1. Prove that

   \[\|A\| = \sup _{x \neq 0} \frac {|\langle Ax, x\rangle |}{\|x\|^2}.\]
2. Prove that at least one of \(\|A\|\) or \(-\|A\|\) is an element of \(\sigma (A)\).

Let \(H\) be a Hilbert space and \(T : H \to H\) be a compact operator. Show that the Hilbert adjoint \(T^*\) is also compact.

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?

Consider the operator \(T:L^2[0,1]\to L^2[0,1]\) defined by \((Tx)(t)=tx(t)\).

1. Show that \(T\) is self-adjoint and satisfies \(\langle x,Tx\rangle \ge 0\) for every \(x\in L^2[0,1]\).
2. Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuum spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\) and resolvent \(\rho (T)\).

The right-shift operator \(R:\ell ^2\to \ell ^2\) is defined by \(R(\xi _1,\xi _2,\ldots )=(0,\xi _1,\ldots )\) and the left-shift operator \(L:\ell ^2\to \ell ^2\) is defined by \(L(\xi _1,\xi _2,\ldots )=(\xi _2,\xi _3,\ldots )\).

1. Show that \(L\) and \(R\) are adjoint to each other.
2. Show that \(\lVert R\rVert =\lVert L\rVert =1\).
3. Prove that the spectrum of \(L\) is given by \[ \sigma _p(L)=\{\lambda \in \mathbb {C}:\lvert \lambda \rvert <1\},\quad \sigma _c(L)=\{\lambda \in \mathbb {C}:\lvert \lambda \rvert =1\},\quad \sigma _r(L)=\emptyset . \]
4. Prove that the spectrum of \(R\) is given by \[ \sigma _p(R)=\emptyset ,\quad \sigma _c(R)=\{\lambda \in \mathbb {C}:\lvert \lambda \rvert =1\},\quad \sigma _r(R)=\{\lambda \in \mathbb {C}:\lvert \lambda \rvert <1\}. \]

Consider the differential operator acting on \(L^2(\mathbb {R})\), \[ L=-\frac {d^2}{dx^2},\qquad 0\le x<\infty \] with self-adjoint boundary conditions \(\psi (0)/\psi '(0)=\tan \theta \) for some fixed angle \(\theta \).

1. Show that when \(\tan \theta <0\) there is a single negative eigenvalue with a normalizable eigenfunction \(\psi _0(x)\) localized near the origin, but none when \(\tan \theta >0\).
2. Show that there is a continuum of eigenvalues \(\lambda =k^2\) with eigenfunctions \(\psi _k(x)=\sin (kx+\eta (k))\), where the phase shift \(\eta \) is found from \[ e^{i\eta (k)}=\frac {1+ik\tan \theta }{\sqrt {1+k^2\tan ^2\theta }}. \]
3. Evaluate the integral \[ I(x,x')=\frac {2}{\pi }\int _0^\infty \sin (kx+\eta (k))\sin (kx'+\eta (k))\dd k, \] and interpret the result with regards the relationship to the Dirac Delta function and completeness, that is, \(\delta (x-x')-I(x,x')=\psi _0(x)\psi _0(x')\). You will need the following standard integral \[ \int _{-\infty }^{\infty } e^{ikx}\frac {1}{1+k^2t^2}\frac {\dd k}{2\pi } =\frac {1}{2\abs {t}}e^{-\abs {x/t}}. \] HINT: you should monitor how the bound state contribution (for \(\tan \theta <0\)) switches on and off as \(\theta \) is varied. Keeping track of the modulus signs \(|\ldots |\) in the standard integral is crucial for this.

Let \(T:H\to H\) be a compact, self-adjoint linear operator defined on a separable Hilbert space \(H\). Let \((v_n)\) be a complete orthonormal sequence of eigenvectors of \(T\) and let \((\lambda _n)\) be the corresponding eigenvalues.

1. Let \(\lambda \in \rho (T)\). Show that the resolvent of \(T\) at \(\lambda \), acting on some \(x\in H\) can be written as \[ R_\lambda x=\sum _{n=1}^{\infty }\frac {1}{\lambda _n-\lambda }\langle x,v_n\rangle v_n. \]
2. Use part (a) to show that if \(\lambda ,\mu \in \rho (T)\) then the first resolvent identity holds: \[ R_\lambda -R_\mu =(\lambda -\mu )R_\lambda R_\mu . \]

Consider the differential operator acting on \(L^2(\mathbb {R})\), \[ L=-\frac {d^2}{dx^2},\qquad 0\le x<\infty \] with self-adjoint boundary conditions \(\psi (0)/\psi '(0)=\tan \theta \) for some fixed angle \(\theta \).

1. Show that when \(\tan \theta <0\) there is a single negative eigenvalue with a normalizable eigenfunction \(\psi _0(x)\) localized near the origin, but none when \(\tan \theta >0\).
2. Show that there is a continuum of eigenvalues \(\lambda =k^2\) with eigenfunctions \(\psi _k(x)=\sin (kx+\eta (k))\), where the phase shift \(\eta \) is found from \[ e^{i\eta (k)}=\frac {1+ik\tan \theta }{\sqrt {1+k^2\tan ^2\theta }}. \]
3. Evaluate the integral \[ I(x,x')=\frac {2}{\pi }\int _0^\infty \sin (kx+\eta (k))\sin (kx'+\eta (k))\dd k, \] and interpret the result with regards the relationship to the Dirac Delta function and completeness, that is, \(\delta (x-x')-I(x,x')=\psi _0(x)\psi _0(x')\). You will need the following standard integral \[ \int _{-\infty }^{\infty } e^{ikx}\frac {1}{1+k^2t^2}\frac {\dd k}{2\pi } =\frac {1}{2\abs {t}}e^{-\abs {x/t}}. \]

HINT: you should monitor how the bound state contribution (for \(\tan \theta <0\)) switches on and off as \(\theta \) is varied. Keeping track of the modulus signs \(\abs {\cdots }\) in the standard integral is crucial for this.

Assume a bounded, self-adjoint operator \(T:H\to H\), where \(H\) is a Hilbert space, satisfies \[ \langle Tx,x\rangle \ge \beta \lVert x\rVert ^2, \tag {1} \] for some \(\beta >0\). The goal of this problem is to show that \(T\) is one-to-one and onto.

1. Show that the nullspace of \(T\) is trivial, i.e. \(\mathcal {N}(T)=\{0\}\).
2. Show that the range \(\mathcal {R}(T)\) is closed.
3. Show that \(\mathcal {R}(T)^\perp =\{0\}\).
4. Why do we have \(\mathcal {R}(T)=H\)?

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 \(Lu=1\).

Define the operator \(K\) on \(L^2[0,1]\) by \[ Ku(x)=\int _0^1 k(x,y)u(y)\,dy, \] for \(x\in [0,1]\), where \(k(x,y)\) is a real-valued, bounded, continuous function, with \(k(x,y)>0\).

1. Find the \(L^2\) adjoint \(K^*\) for \(K\), and find a condition on \(k\) which guarantees that \(K\) is self-adjoint.
2. Describe the properties of the set \[ \{\lambda \in \mathbb {C}:(K-\lambda I)u=0\text { for some }u\ne 0\} \] (cardinality, bounds, etc..).
3. Consider the problem \(Ku=f\). Let \(X\) be the set of all \(f\in L^2[0,1]\) such that a unique solution \(u=K^{-1}f\) exists. Can we be assured that there is a constant \(C\) such that \(\lVert K^{-1}f\rVert \le C\lVert f\rVert \), for all \(f\in X\)? Why or why not?

State and prove the Riesz Representation Theorem for a Hilbert space.

For the differential operator \(Lu=u''+(x^2-1)u'\) on the interval \([0,1]\) with boundary conditions \(u(0)=u'(1)\) and \(u(1)=u'(0)\) find the adjoint operator and its domain.

1. Find the first term of the asymptotic representation of \[ n!=\int _0^\infty \exp (-t)t^{a-1}\,dt \] for large \(n\). (This approximation is called Stirling’s formula.)
2. What is the order of the error term and how can this be rigorously justified? (Invoke the appropriate theorem.)

1. State the Riesz Representation Theorem (a proof is not required).
2. Use the Riesz Representation theorem to prove that a bounded linear operator on a Hilbert Space has an adjoint. Prove that the adjoint is a bounded linear operator on the Hilbert space.
3. Find the adjoint for the operator \[ (Ku)(y)=\int _0^1 \min (x,y)u(x)\,dx. \]

1. Prove that the eigenvalues of a self-adjoint linear operator acting in a Hilbert space \(H\) are real and that the eigenfunctions of distinct eigenvalues are orthogonal.
2. Prove that if \(L\) is a bounded linear operator in \(H\) then the adjoint \(L^*\) is also a bounded linear operator.

Prove that if \(L\) is a bounded linear operator in a Hilbert Space \(H\), then the adjoint operator \(L^*\) exists.

Consider the non-local differential operator \[ Lu=Du''-u(g(x)) \] with boundary conditions \(u(0)=u(1)=0\) and where \(g(x)\) is a monotone increasing function with \(g(0)=0\), \(g(1)=1\), and \(g(x)<x\) for \(0<x<1\).

1. Find the adjoint of the operator \(L\).
2. Prove that the operator \(L\) is invertible.
3. Devise an iteration scheme that could be used to solve \(Lu=f(x)\). Under what conditions on \(D\) is this iteration scheme a contraction mapping?

Use piecewise linear finite element basis functions with uniform grid size \(h\) and the Galerkin method to find an approximate representation for the differential equation \(a\frac {d^2u}{dx^2}+b\frac {du}{dx}+u(x)=f(x)\), with \(u(0)=0\), \(u(1)=0\), on the interval \([0,1]\). (It is only necessary to find the approximating equations, not to solve them.)

Consider the differential operator \(Lu=u''+k^2u\), \(k\ne 0\), subject to homogeneous boundary conditions \(u(0)=u(1)=0\). Show by direct calculation that the associated Green’s function is \[ g(x,y)= \begin {cases} \dfrac {\sin kx\,\sin k(y-1)}{k\sin k}, & 0\le x<y\le 1,\\[1.2ex] \dfrac {\sin ky\,\sin k(x-1)}{k\sin k}, & 0\le y<x\le 1, \end {cases} \] provided \(\sin k\ne 0\). Using the fact that \(L\) is self-adjoint, show that the solution to the equation \[ u''(x)+k^2u(x)=f(x) \] with inhomogeneous boundary data \(u(0)=\alpha \), \(u(1)=\beta \) is \[ u(x)=\int _0^1 g(x,y)f(y)\,dy-\frac {\alpha }{\sin k}\sin k(x-1) +\frac {\beta }{\sin k}\sin kx \] provided \(\sin k\ne 0\). What happens when \(k=m\pi \)?

1. State the Riesz Representation Theorem.
2. Use the Riesz Representation theorem to prove that a bounded linear operator on a Hilbert Space has an adjoint. Prove that the adjoint is a bounded linear operator on the Hilbert space.
3. Find the adjoint for the operator \[ (Ku)(y)=\int _0^1 \min (x,y)u(x)\,dx. \]

1. Find the Fourier transform of \(\dfrac {\sin x}{x}\).
2. Suppose that the Fourier transform of the function \(f(t)\) is 0 for frequencies whose absolute value is greater than \(\pi /h\). Find the representation of \(f(t)\) in terms of sinc functions.

Consider the problem \[ \begin {aligned} Lu &\equiv -\frac {d}{dx}\left (p\frac {du}{dx}\right )+qu = \lambda u,\\ \alpha u(0)+\beta \frac {du}{dx}(0) &= 0,\\ \gamma u(1)+\delta \frac {du}{dx}(1) &= 0 \end {aligned} \tag {5.1} \] where \(x\in [0,1]\), \(p\in C^1[0,1]\), \(q\in C[0,1]\), and \(\alpha ,\beta ,\gamma ,\delta \) are constants.

1. Show that (5.1) is self-adjoint if and only if \(p\) and \(q\) are real, \(\gamma \bar {\delta }=\bar {\gamma }\delta \), and \(\alpha \bar {\beta }=\bar {\alpha }\beta \).
2. Suppose hereafter that \(p=q=1\), \(\alpha =1\), \(\beta =0\), \(\gamma =0\), \(\delta =1\). Compute the eigenvalues and eigenfunctions for \(L\) with the given boundary conditions.
3. Compute the Green’s function for \(L\) (with coefficients and boundary conditions as in (b)).
4. Show how to solve the nonhomogeneous problem \[ Lu=f(x) \] with \(L\) and the boundary conditions as in (c). What conditions on \(f\) suffice to make \(u\) a classical solution?

Let \(H\) be a Hilbert space and \(L:H\to H\).

1. Find and prove the condition that ensures \[ Lu=\int _0^1 k(x,y)u(y)\,dy \] is a bounded linear operator in \(L^2[0,1]\).
2. Under what condition does the adjoint operator \(L^*\) exist? Assuming that this condition is satisfied, give \(L^*\).

A bounded linear operator \(P\) in a complex Hilbert space \(\mathcal {H}\) is a projection if and only if \(P^2=P\). Prove that a projection \(P\) is an orthogonal projection (so that \(P\mathcal {H}\perp (1-P)\mathcal {H}\)) if and only if \(P\) is selfadjoint. (You may use the known theorem that a bounded linear operator \(A\) in \(\mathcal {H}\) is selfadjoint if and only if \((Ax,x)\) is real for all \(x\in \mathcal {H}\).)

Define a regular Sturm-Liouville operator by the equation \[ \mathcal {L} u=-(2x)^{-1}\bigl ((2x)^{-1}u'\bigr )' \] on \(1\le x\le \sqrt {2}\) using boundary conditions \(u(1)=u(\sqrt {2})=0\).

1. Explain how \(\mathcal {L}\) and the boundary conditions can be used to construct a selfadjoint operator \(S\) in a suitable Hilbert space \(\mathcal {H}\). Do not give proofs.
2. Calculate the eigenvalues and eigenfunctions of \(S\) using the solution basis \(u_1(x)=\sin (\sqrt {\lambda }x^2)\), \(u_2(x)=\cos (\sqrt {\lambda }x^2)\) of the differential equation \(\mathcal {L} u-\lambda u=0\). Do not derive the basis.
3. Verify by direct integration that the eigenfunctions are orthogonal in \(\mathcal {H}\).

Let \(S\) be the selfadjoint realization in the Hilbert space \(\mathcal {H}=L_2(0,\infty )\) for the singular Sturm-Liouville operator \(\mathcal {L}=-u''\) on \(0<x<\infty \) with boundary condition \(u'(0)=0\). Assume that \(S\) is defined by the domain \[ D(S)=\{u\in \mathcal {H}:u''\in \mathcal {H}\text { and }u'(0)=0\} \] and the relation \(Su=-u''\) for all \(u\in D(S)\).

1. Show that the Green’s function \(G\) for the operator \(S\) is given by the equation below, where \(\zeta =re^{i\theta }\) and \(\sqrt {\zeta }=\sqrt {r}e^{i\theta /2}\) on \(0<\theta <2\pi \): \[ G(x,y,\zeta )= \begin {cases} i e^{i\sqrt {\zeta }y}\cos (\sqrt {\zeta }x)/\sqrt {\zeta }, & 0\le x\le y<\infty ,\\ i e^{i\sqrt {\zeta }x}\cos (\sqrt {\zeta }y)/\sqrt {\zeta }, & 0\le y\le x<\infty . \end {cases} \]
2. Prove that the operator \(S\) has a pure continuous spectrum consisting of all non-negative real numbers.

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.

Using an integral representation for the \(n\)-th order Bessel function \(J_n(x)\), find the leading order asymptotic representation of \(J_n(x)\) for large positive \(x\).

State and prove the Classical Fourier pointwise convergence theorem for the representation of a function by its Fourier series.

Define a regular Sturm-Liouville operator by the equation \[ \mathcal {L} u=-x^2(x^2u')' \] on \(1/2\le x\le 1\) using boundary conditions \(u(1/2)=u(1)=0\). Complete the following:

1. Explain how \(\mathcal {L}\) and the boundary conditions can be used to construct a selfadjoint operator \(S\) in a suitable Hilbert space \(\mathcal {H}\). Do not give proofs.
2. Calculate the eigenvalues and eigenfunctions of the selfadjoint operator \(S\), using the solution basis \(u_1(x)=\sin (\lambda /x)\), \(u_2(x)=\cos (\lambda /x)\) of the differential equation \[ x^2(x^2u')'+\lambda u=0. \] Do not derive the basis.
3. Verify by direct integration that the eigenfunctions are orthogonal in the Hilbert space \(\mathcal {H}\).

Let \(S\) be the selfadjoint realization in the Hilbert space \(\mathcal {H}=L_2(0,\infty )\) for the singular Sturm-Liouville operator \(\mathcal {L}=-u''\) on \(0<x<\infty \) with boundary condition \(u(0)=0\). Assume that \(S\) is defined by the domain \[ D(S)=\{u\in \mathcal {H}:u''\in \mathcal {H}\text { and }u(0)=0\} \] and the relation \(Su=-u''\) for all \(u\in D(S)\). Complete the following:

1. Show that the Green’s function \(G\) for the operator \(S\) is given by the equation below, where \(\xi =re^{i\theta }\) and \(\sqrt {\xi }=\sqrt {r}e^{i\theta /2}\) on \(0<\theta <2\pi \). \[ G(x,y,\xi )= \begin {cases} e^{i\sqrt {\xi }y}\sin (\sqrt {\xi }x)/\sqrt {\xi } & \text {for }0\le x\le y<\infty ,\\ e^{i\sqrt {\xi }x}\sin (\sqrt {\xi }y)/\sqrt {\xi } & \text {for }0\le y\le x<\infty . \end {cases} \]
2. Prove that the operator \(S\) has a pure continuous spectrum consisting of all non-negative real numbers.

Let \(T\) be a linear operator in a Hilbert space \(\mathcal {H}\) with domain \(D(T)\) dense in \(\mathcal {H}\). Complete the following:

1. Define the adjoint operator \(T^*\).
2. Let \(G(T)\) be the graph of \(T\) in \(\mathcal {H}\oplus \mathcal {H}\). Define the operator \(V\) on \(\mathcal {H}\oplus \mathcal {H}\) by \(V(f,g)=(g,-f)\). Prove that the graph of \(T^*\) is the orthogonal complement in \(\mathcal {H}\oplus \mathcal {H}\) of \(V(G(T))\), that is, \[ G(T^*)=(V(G(T)))^\perp . \]
3. Let \(\overline {G(T)}\) be the closure in \(\mathcal {H}\oplus \mathcal {H}\) of the graph of \(T\). Prove that \[ \mathcal {H}\oplus \mathcal {H}=V(\overline {G(T)})\oplus G(T^*). \]

Find the adjoint operator, its domain and the least squares solution of the problem \(Lu\equiv 4u''+u=e^x\) for \(0\le x\le \pi /2\) if \(u(0)=\alpha \) and \(u(\pi /2)=\beta \).

Let \(A\) be a selfadjoint operator in a Hilbert space \(\mathcal {H}\) and let \(\{E_\lambda \}\) be the spectral family of \(A\). Prove that if a real number \(\mu \) is in the resolvent set of \(A\) then there exists an interval \(I=\{\lambda :\lvert \lambda -\mu \rvert <\delta \}\) such that \(E_\lambda =\text {const.}\) for all \(\lambda \in I\). (You may use without proof the Stieltjes-Hellinger-Stone theorem: \[ R_z=(A-z)^{-1}. \] \[ \bigl ((E_{b+}+E_{b-}-E_{a+}-E_{a-})u,u\bigr ) =\lim _{\epsilon \to 0+}\frac {1}{\pi i} \int _a^b\bigl ((R_{\lambda +i\epsilon }-R_{\lambda -i\epsilon })u,u\bigr )\,d\lambda . \]

Consider the regular Sturm-Liouville operator \[ \mathcal {L} u=-e^{-2x}(e^{2x}u')'-u \] on \(0\le x\le \pi \) with boundary conditions \(u(0)=0\), \(u(\pi )=0\).

1. Show how \(\mathcal {L} u\) and the boundary conditions can be used to construct a selfadjoint operator \(S\) in a suitable Hilbert space \(\mathcal {H}\).
2. Calculate the eigenvalues and the corresponding eigenfunctions of \(S\).

The singular Sturm-Liouville operator \(Lu=-u''\) on \(-\infty <x<\infty \) is known to have a unique selfadjoint realization \(T\) in the Hilbert space \(H=L_2(-\infty ,\infty )\). It is defined by \(D(T)=H\cap \{u:u''\in H\}\) and \(Lu=-u''\) for all \(u\in D(T)\). (Do not prove this). Calculate the Green’s function (resolvent kernel) of \(T\).