Point, Continuous, and Residual Spectrum

Consider the operator \(T : L^2[0, 1] \to L^2[0, 1]\) defined by

Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuum spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\) and resolvent set \(\rho (T)\).

Let \(X\) be a normed space over \(\mathbb {R}\) and \(Y \subset X\) be a linear subspace. For \(x_0 \in X \setminus Y\) we define the linear functional

by

1. Prove that \(f\) is continuous if and only if \(x_0 \notin \overline {Y}\).

2. Prove that \(\overline {Y} = \bigcap \{\operatorname {Ker} g \mid g \in X', Y \subset \operatorname {Ker} g\}\), where \(X'\) is the dual of \(X\).

   Hint for part \((ii)\): The \(\subset \) direction is relatively easy. For the other direction, note that as given in part (i), \(f\) satisfies \(f|_Y = 0\) and \(f(x_0) = 1\). The Hahn-Banach theorem provides an extension of \(f\) to the entire space, use this to prove the other direction by contradiction.

Let \((\lambda _j)\) be a sequence that is dense in \([0, 1]\) and consider the linear operator \(T : \ell ^2 \to \ell ^2\) defined by its action on a sequence \((\xi _j) \in \ell ^2\) by

Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuum spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\) and resolvent \(\rho (T)\).

Let \((\lambda _j)\) be a sequence that is dense in \([0, 1]\) and consider the linear operator \(T : \ell ^2 \to \ell ^2\) defined by its action on a sequence \((\xi _j) \in \ell ^2\) by

Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuum spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\) and resolvent \(\rho (T)\).

Consider the left shift operator \(L : \ell ^2 \to \ell ^2\), defined for a sequence \((x_n) \in \ell ^2\) by

1. Prove that \(\|L\| = 1\).
2. Explain why if \(\lambda \in \mathbb {C}\) with \(|\lambda | > 1\), then \(\lambda \in \rho (L)\), the resolvent set of \(L\).
3. Prove that that if \(\lambda \in \mathbb {C}\) with \(|\lambda | < 1\), then \(\lambda \in \sigma _p(L)\), the point spectrum of \(L\).
4. Deduce what is \(\sigma (L)\), the spectrum of \(L\).

Show that a mapping \(T:X\to Y\) where \(X\) and \(Y\) are metric spaces is continuous if and only if the inverse image of any closed set \(M\subset Y\) is a closed set in \(X\).

Consider the left shift operator \(L:\ell ^2\to \ell ^2\), defined for a sequence \((x_n)\in \ell ^2\) by \[ L(x_1,x_2,x_3,\ldots )=(x_2,x_3,x_4,\ldots ). \]

1. Prove that \(\lVert L\rVert =1\).
2. Explain why if \(\lambda \in \mathbb {C}\) with \(\lvert \lambda \rvert >1\), then \(\lambda \in \rho (L)\), the resolvent set of \(L\).
3. Prove that that if \(\lambda \in \mathbb {C}\) with \(\lvert \lambda \rvert <1\), then \(\lambda \in \sigma _p(L)\), the point spectrum of \(L\).
4. Deduce what is \(\sigma (L)\), the spectrum of \(L\).

The small longitudinal free vibrations of an elastic bar are governed by the following equation: \[ \rho (x)\frac {\partial ^2u}{\partial t^2} =\frac {\partial }{\partial x}\left [E(x)\frac {\partial u}{\partial x}\right ]. \] Here \(u\) is the longitudinal displacement \(\rho \) is the linear density of the material, and \(E\) is its Young’s modulus. Assume that the bar is constructed by welding together two bars of different (constant) Young’s moduli \(E_1\), \(E_2\) and densities \(\rho _1\), \(\rho _2\), respectively. The displacement \(u\) is continuous across the junction, which is located at \(x=0\).

1. Give a weak formulation of the global initial value problem, and use this to derive the following jump condition: \[ E_1u_x(0-,t)=E_2u_x(0+,t),\qquad t>0. \]
2. Let \(c_j^2=E_j/\rho _j\). A left incoming wave \(u_I(x,t)=e^{i(t-x/c_1)}\) produces at the junction a reflected wave \(u_R(x,t)=Re^{i(t+x/c_1)}\) and a transmitted wave \(u_T=Te^{i(t-x/c_2)}\). Determine the reflection and transmission coefficients \(R,T\), and interpret the result.

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 \(\rho (T)\).

\(F(\omega )\) is the Fourier transform of \(f(x)\) (which is continuous and not identical zero), \(x\) and \(\omega \) are real variables. Explain why it is impossible that both functions \(f(x)\) and \(F(\omega )\) have finite support.

The small longitudinal free vibrations of an elastic bar are governed by the following equation: \[ \rho (x)\frac {\partial ^2u}{\partial t^2} =\frac {\partial }{\partial x}\left [E(x)\frac {\partial u}{\partial x}\right ]. \] Here \(u\) is the longitudinal displacement, \(\rho \) is the linear density of the material and \(E\) is its Young’s modulus. Assume that the bar is constructed by welding together two bars of different (constant) Young’s moduli \(E_1\), \(E_2\) and densities \(\rho _1\), \(\rho _2\), respectively. The displacement \(u\) is continuous across the junction, which is located at \(x=0\).

1. Give a weak formulation of the global initial value problem, and use this to derive the following jump condition: \[ E_1u_x(0-,t)=E_2u_x(0+,t),\qquad t>0. \]
2. Let \(c_j^2=E_j/\rho _j\). A left incoming wave \(u_I(x,t)=e^{i(t-x/c_1)}\) produces at the junction a reflected wave \(u_R(x,t)=Re^{i(t+x/c_1)}\) and a transmitted wave \(u_T=Te^{i(t-x/c_2)}\). Determine the reflection and transmission coefficients \(R,T\), and interpret the result.

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

1. Show that \(T\) is a bounded, self-adjoint linear operator.
2. Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuous spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\), and resolvent \(\rho (T)\).
3. Show that the corresponding spectral family \((E_\lambda )_{\lambda \in \mathbb {R}}\) is given by \[ E_\lambda x= \begin {cases} 0, & \lambda <0,\\ v_\lambda , & 0\le \lambda \le 1,\\ x, & \lambda >1, \end {cases} \qquad \text {where}\qquad v_\lambda (t)= \begin {cases} x(t), & 0\le t\le \lambda ,\\ 0, & \lambda <t\le 1. \end {cases} \]

The small longitudinal free vibrations of an elastic bar are governed by the following equation: \[ \rho (x)\frac {\partial ^2u}{\partial t^2} =\frac {\partial }{\partial x}\left [E(x)\frac {\partial u}{\partial x}\right ]. \] Here \(u\) is the longitudinal displacement, \(\rho \) is the linear density of the material, and \(E\) is its Young’s modulus. Assume that the bar is constructed by welding together two bars of different (constant) Young’s moduli \(E_1,E_2\) and densities \(\rho _1,\rho _2\), respectively. The displacement \(u\) is continuous across the junction, which is located at \(x=0\).

1. Give a weak formulation of the global initial value problem, and use this to derive the following jump condition: \[ E_1u_x(0-,t)=E_2u_x(0+,t),\qquad t>0. \]
2. Let \(c_j^2=E_j/\rho _j\). A left incoming wave \(u_I(x,t)=e^{i(t-x/c_1)}\) produces at the junction a reflected wave \(u_R(x,t)=Re^{i(t+x/c_1)}\) and a transmitted wave \(u_T=Te^{i(t-x/c_2)}\). Determine the reflection and transmission coefficients \(R,T\), and interpret the result.

Let \((\lambda _j)\) be a sequence real numbers with \(\lambda _j\ne 1\) for all \(j\) and \(\lambda _j\to 1\). Consider the operator \(T:\ell ^2\to \ell ^2\) defined for \((\xi _j)\in \ell ^2\) by \[ T(\xi _j)=(\lambda _j\xi _j). \]

1. Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuum spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\) and resolvent set \(\rho (T)\).
2. Give a condition on the \(\lambda _j\) for \(T\) to be invertible.

Let \(D_1\) and \(D_2\) be two disjoint domains, whose boundaries share a common curve \(\Gamma \). Let

• \(f(z)\) be analytic in \(D_1\) and continuous in \(D_1\cup \Gamma \)
• \(g(z)\) be analytic in \(D_2\) and continuous in \(D_2\cup \Gamma \)
• \(f(z)=g(z)\) for \(z\in \Gamma \)

Show that the function \[ H(z)= \begin {cases} f(z), & z\in D_1,\\ f(z)=g(z), & z\in \Gamma ,\\ g(z), & z\in D_2 \end {cases} \] is analytic in \(D=D_1\cup \Gamma \cup D_2\).

[Hint: Use the Morera theorem.]

The small longitudinal free vibrations of an elastic bar are governed by the following equation: \[ \rho (x)\frac {\partial ^2u}{\partial t^2} =\frac {\partial }{\partial x}\left [E(x)\frac {\partial u}{\partial x}\right ]. \] Here \(u\) is the longitudinal displacement, \(\rho \) is the linear density of the material and \(E\) is its Young’s modulus. Assume that the bar is constructed by welding together two bars of different (constant) Young’s moduli \(E_1,E_2\) and densities \(\rho _1,\rho _2\), respectively. The displacement \(u\) is continuous across the junction, which is located at \(x=0\).

1. Give a weak formulation of the global initial value problem, and use this to derive the following jump condition: \[ E_1u_x(0-,t)=E_2u_x(0+,t),\qquad t>0. \]
2. Let \(c_j^2=E_j/\rho _j\). A left incoming wave \(u_I(x,t)=e^{i(t-x/c_1)}\) produces at the junction a reflected wave \(u_R(x,t)=R e^{i(t+x/c_1)}\) and a transmitted wave \(u_T=T e^{i(t-x/c_2)}\). Determine the reflection and transmission coefficients \(R,T\), and interpret the result.

Let \(f(x,t)\) be continuous for all \((x,t)\in \mathbb {R}^n\times \mathbb {R}\) and let \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Suppose that there is \(M\in \mathbb {R}\) such that \[ \norm {f(x,t)-f(y,t)}\le M\norm {x-y}, \qquad \text {for all }(x,t),(y,t)\in \mathbb {R}^n\times \mathbb {R}. \] Show that there is and \(\epsilon >0\) and a unique function \(y\in C^1([0,\epsilon ],\mathbb {R}^n)\) that satisfies the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(0)=x_0. \end {cases} \]

The small longitudinal free vibrations of an elastic bar are governed by the following equation: \[ \rho (x)\frac {\partial ^2u}{\partial t^2} =\frac {\partial }{\partial x}\left [E(x)\frac {\partial u}{\partial x}\right ]. \] Here \(u\) is the longitudinal displacement, \(\rho \) is the linear density of the material and \(E\) is its Young’s modulus. Assume that the bar is constructed by welding together two bars of different (constant) Young’s moduli \(E_1,E_2\) and densities \(\rho _1,\rho _2\), respectively. The displacement \(u\) is continuous across the junction, which is located at \(x=0\).

1. Give a weak formulation of the global initial value problem, and use this to derive the following jump condition: \[ E_1u_x(0-,t)=E_2u_x(0+,t),\qquad t>0. \]
2. Let \(c_j^2=E_j/\rho _j\). A left incoming wave \(u_I(x,t)=e^{i(t-x/c_1)}\) produces at the junction a reflected wave \(u_R(x,t)=R e^{i(t+x/c_1)}\) and a transmitted wave \(u_T=T e^{i(t-x/c_2)}\). Determine the reflection and transmission coefficients \(R,T\), and interpret the result.

Let \(f(t,x)\) be continuous function on \(\mathbb {R}\times \mathbb {R}^n\) that satisfies \[ \norm {f(t,x_1)-f(t,x_2)}\le \norm {x_2-x_1} \] for all \(t\in \mathbb {R}\) and \(x_1,x_2\in \mathbb {R}^n\). Let \(x_0\in \mathbb {R}^n\) be a point. Show that there is and \(\epsilon >0\) and a unique function \(y\in C^1([0,\epsilon ],\mathbb {R}^n)\) that satisfies the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(0)=x_0. \end {cases} \]

Let \(f(x,t)\) be continuous for \((x,t)\in \mathbb {R}^n\times \mathbb {R}\) and let \(x_0\in \mathbb {R}^n\) be a point. Show that there is and \(\epsilon >0\) and a function \(y\in C^1([0,\epsilon ],\mathbb {R}^n)\) that satisfies the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(0)=x_0. \end {cases} \]

Suppose \(f(x,t)\) is continuous for \((x,t)\in \mathbb {R}^n\times \mathbb {R}\) and let \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\) be a point. Assume that there is a constant \(L<\infty \) so that \[ \lvert f(x_1,t)-f(x_2,t)\rvert \le L\lvert x_1-x_2\rvert \] for all \(x_1,x_2,t\). Show that there is and \(\epsilon >0\) and a unique function \(y\in C^1([t_0,t_0+\epsilon ],\mathbb {R}^n)\) that satisfies the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(t_0)=x_0. \end {cases} \]

1. Define Cauchy sequence, and what it means for a linear space to be complete.
2. Let \(C[0,1]\) be the set of real-valued functions which are continuous on \([0,1]\). Show that \(C[0,1]\) is complete under under the uniform norm \(\lVert f\rVert =\sup _{x\in [0,1]}|f(x)|\).
3. Let \(f_n(x)=x^n\) for \(x\in [0,1]\). Find \(f=\lim _{n\to \infty }f_n\). Is \(f(x)\) continuous? Does this violate the completeness you showed in (a)? Explain.

1. Explain what is meant by a Cauchy sequence, Completeness of a space, and a Hilbert space (you can assume the definitions of norm, inner product, and linear space are known).
2. Prove that the set of continuous functions on the interval \([a,b]\) is complete with respect to the uniform norm \[ \lVert f\rVert =\max _{x\in [a,b]} |f(x)|. \]

Consider the system of ordinary differential equations \[ u'=-\lvert u\rvert ^2u+h(t), \tag {1} \] where \[ h:[0,1]\to \mathbb {R}^N \] is a continuous function and \(\lvert \cdot \rvert \) is the Euclidean norm in \(\mathbb {R}^N\). Use Brouwer’s fixed point theorem to show that (1) has a solution \(u\in C^1[0,1]\) such that \[ u(0)=u(1). \] Use this result to conclude that if \[ h:\mathbb {R}\to \mathbb {R}^N \] is continuous and periodic of period 1, then (1) has a solution of period 1.

(Hint: Consider equation (1) subject to initial data \(u_0:\lvert u_0\rvert \le r\), where \(r\) is large (in relation to \(h\)) and examine solution curves by computing \(\frac {d}{dt}\lvert u\rvert ^2\) for solutions \(u\).) In relation to the above, what can you say about the equation \[ u'=\lvert u\rvert ^2u+h(t). \tag {2} \]

Suppose the function \(\phi (x)\) is Lipschitz continuous on the unit interval \(0<x<2\pi \) and that \(\lim _{x\to 0}\phi (x)=\phi (0^+)\) exists. Suppose \(\int _0^\infty \frac {\sin x}{x}\,dx=\alpha \) is known. Evaluate \(\lim _{k\to \infty }\int _0^{2\pi }\phi (x)\frac {\sin kx}{x}\,dx\) (with \(k\) restricted to be integer-valued).

Let \(G\) be an open subset of \(\mathbb {R}^N\) and let \(f\in L^2(G)\). Show that \(f\) defines a continuous linear functional on \(H^1(G)\).

Let \(\zeta \in C_0(\mathbb {R})\) be continuous with compact support such that the integral \[ \int _{-\infty }^{\infty }\zeta (x)\,dx=1. \] Show that as distributions, the convolutions \(\zeta _\epsilon *F\to F\) as \(\epsilon \to 0\), where \[ \zeta _\epsilon (x):=\frac {1}{\epsilon }\zeta \left (\frac {x}{\epsilon }\right ). \]

Suppose \(D\) is the rectangle defined by \(\lvert x-x_0\rvert \le a\), \(\lvert y-y_0\rvert \le b\) in the plane \(\mathbb {R}^2\). Suppose that \(f\) is a real valued function defined on \(D\) which is continuous, hence bounded by some nonnegative number \(M\) on \(D\).

1. Show that if \(\alpha =\min \{a,b/M\}\), then there exists a function \(y(x)\), defined on \(\lvert x-x_0\rvert \le \alpha \), such that \[ y'(x)=f(x,y(x))\quad \text {and}\quad y(x_0)=y_0. \]
2. Show that the solutions \(y\) guaranteed in part (i) may not be unique; that is, give an example in which (i) holds and more than one solution \(y(x)\) exists through the point \((x_0,y_0)\) in your example.

Define the concepts “strongly continuous semigroup on \(H\)” and “infinitesimal generator” of the semigroup.

Let \(\{T(t):t\ge 0\}\) be a strongly continuous semigroup on \(H\) with infinitesimal generator \(A\). Show that \[ T(t):D(A)\longrightarrow D(A) \] and that the map \[ t\longmapsto T(t)x,\qquad t\in [0,\infty ), \] is differentiable and \[ \frac {d}{dt}T(t)x=AT(t)x=T(t)Ax,\quad x\in D(A). \]