Bounded Linear Operators

Let \(T : X \to X\) be a compact linear operator on a normed space \(X\). Show that for every \(\lambda \neq 0\) the null space \(N(T - \lambda I)\) is finite dimensional.

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 \(V,W\) be Banach spaces, and let \(T_i:V\to W\) be a sequence of continuous linear operators such that \(\lim _{i\to \infty }T_i v\) exists for every \(v\in V\). Define the linear map \(T:V\to W\) by \[ Tv=\lim _{i\to \infty }T_i v \]

Show that \(T\) is continuous.

Let \(X\) be a Banach space and \(A\), \(T\) be bounded linear operators from \(X\) to itself. Assume further that \(A\) is compact. Show that \(TA\) and \(AT\) are also compact operators.

For \(k \in \mathbb {N}\) let \(T_k : \ell ^1 \to \ell ^\infty \) be the linear operator given by

where \(e_j = (0, 0, \ldots , 1, \ldots , 0, 0, \ldots )\), i.e. it is the element of \(\ell ^\infty \) with a single 1 in the \(j\)th position and all other entries are zero.

1. Compute \(\|T_k\|\) for each \(k\), and conclude that each \(T_k\) is bounded.
2. Prove that the sequence \((T_n)\) is strongly operator convergent.
3. Prove that the sequence \((T_n)\) is not uniformly operator convergent.

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 \((\lambda _j)\) 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 \(X\) be a Banach space over the complex numbers. Let \(X_{\mathbb {R}}\) be the vector space over the reals obtained by restricting the multiplication by scalars used to define \(X\) to the real line. Let \(f : X \to \mathbb {C}\) be a bounded linear functional on \(X\) and let \(\phi : X \to \mathbb {R}\) be its real part, defined by \(\phi (x) = \operatorname {Re} f(x)\) for all \(x \in X\).

1. Prove that \(\phi \) is a linear functional on \(X_{\mathbb {R}}\).

2. Prove that \(f\) can be reconstructed from \(\phi \) by:

   \[f(x) = \phi (x) - i\phi (ix), \quad \text {for } x \in X.\]

3. Show that \(f\) as a linear functional on \(X\) and \(\phi \) as a linear functional on \(X_{\mathbb {R}}\) have the same norm, i.e.

   \[\|f\|_{X^*} = \|\phi \|_{X_{\mathbb {R}}^*},\]

   where \(X^*\) is the dual of \(X\) and \(X_{\mathbb {R}}^*\) is the dual of \(X_{\mathbb {R}}\). Hint: To prove one of the inequalities, it may be helpful to find \(\lambda \in \mathbb {C}\) such that \(|f(x)| = f(x/\lambda )\) for a fixed \(x\) with \(f(x) \neq 0\).

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 \(H\) be a separable Hilbert space with \((e_n)\) being a total orthonormal sequence of \(H\). Let \(T : H \to H\) be a bounded linear operator satisfying

Show that \(T\) is compact by approximating \(T\) by a sequence of finite rank operators \((T_k)\) that converges to \(T\) in an appropriate sense.

Let \(X\) be a Banach space. Let \(y \in X\) be fixed and let \(f : X \to \mathbb {R}\) be a bounded linear functional. Show that there is a constant \(C > 0\) such that for all scalars \(\alpha \) with \(|\alpha | < C\) the equation

has a unique solution on the open ball \(\{x \in X : \|x - y\| < 1\}\).

Let \(X\) be a real Banach space. Assume \(f \in X^*\) has a closed nullspace \(\mathcal {N}(f)\). The goal of this problem is to show that \(f\) must be a bounded linear functional.

1. Let \(x_0 \in X\) be such that \(f(x_0) = 1\). Explain why there is an \(\epsilon > 0\) for which the ball

   \[B(x_0, \epsilon ) \equiv \{x \in X \mid \|x - x_0\| < \epsilon \}\]

   satisfies \(B(x_0, \epsilon ) \subset X - \mathcal {N}(f)\).

2. Prove that \(f(x) > 0\) for all \(x \in B(x_0, \epsilon )\), where \(\epsilon \) is as in part (a). To prove this, you may assume for contradiction that there is some \(y \in B(x_0, \epsilon )\) with \(f(y) \le 0\).
3. Any \(x \in B(x_0, \epsilon )\) can be written as \(x = x_0 + \epsilon u\) for some \(u\) with \(\|u\| < 1\). Use the result of part (b) to show that \(|f(u)| < 1/\epsilon \).
4. To conclude, give an upper bound for \(\|f\|\).

Let \(H\) be a separable Hilbert space with \((e_n)\) being a total orthonormal sequence of \(H\). Let \(T : H \to H\) be a bounded linear operator satisfying

Show that \(T\) is compact by approximating \(T\) by a sequence \((T_k)\) of finite rank operators that converges to \(T\) in an appropriate sense.

Let \(T : H \to H\) be a bounded linear operator on a Hilbert space \(H\). Show that \(T\) is compact if and only if \(T^*T\) is compact.

Let \(H\) be a separable Hilbert space with \((e_n)\) being a total orthonormal sequence of \(H\). Let \(T : H \to H\) be a bounded linear operator satisfying

Show that \(T\) is compact by approximating \(T\) by a sequence \((T_k)\) of finite rank operators that converges to \(T\) in an appropriate sense.

Let \(V\), \(W\) be Banach spaces, and let \(T_{i} : V \to W\) be a sequence of bounded linear operators such that \(\lim _{i \to \infty } T_{i}v\) exists for every \(v \in V\). Define the linear map \(T : V \to W\) by \[ Tv = \lim _{i \to \infty } T_{i}v. \]

Show that \(T\) is bounded.

Let \(H\) be a Hilbert space endowed with the inner product \(\langle \cdot , \cdot \rangle \). Let \(T\) be a bounded linear operator satisfying \(\langle T x, x\rangle \ge 0\) for all \(x \in H\). Show that \(I + T\) is injective.

Let \(T : H \to H\) be a bounded linear operator on a Hilbert space \(H\). Let \(A\) and \(B\) be subsets of \(H\) such that \(T(A) \subset B\). Show that \(A^\perp \supset T^*(B^\perp )\). Note: We used the notation \(T(A) = \{T x \mid x \in A\}\).

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 \(T : X \to X\) be a bounded linear operator on a complex Banach space \(X\). Prove that \(\sigma (T)\) lies in the complex plane disk: \(\{\lambda \in \mathbb {C} \mid |\lambda | \le \|T\|\}\).

1. State the Closed Graph Theorem.
2. Let \(T : U \to V\) be a linear map between two Banach spaces such that, for any bounded linear functional \(f\) on \(V\) , the composite \(f \circ T\) is a bounded functional on \(U\) . Prove that \(T\) is bounded. You may use the fact that bounded linear functionals on a Banach space separate points.

1. State the Baire Category Theorem.
2. Let \(V\) be a Banach space, and \(T : V \to V\) a bounded linear map. Assume that for every \(v \in V\) there exists a non-negative integer \(n\) such that \(T^{n} (v) = 0\). Prove that there exists an integer \(n\) such that \(T^{n} (v) = 0\) for all \(v \in V\).

Let \(X\) be a normed space and \(X^*\) be the space of bounded linear functionals. Every \(x\in X\) defines a functional \(\hat {x}\) on \(X^*\) by \(\hat {x}(f)=f(x)\) for all \(f\in X^*\). Prove that \(\lVert \hat {x}\rVert =\lVert x\rVert \).

Consider the sequence of linear operators \((U_n)\) where \(U_n:H\to H\), \(H\) is a Hilbert space and the \(U_n\) are unitary, i.e. \(U_n^*U_n=I\), where \(I\) is the identity operator on \(H\). Show that if \(U_n\to U\) in the operator norm, the limit \(U\) must also be unitary, i.e. \(U^*U=I\).

Let \(X\) be a real Banach space, \(f:X\to \mathbb {R}\) be a bounded linear functional and \(T:X\to X\) a bounded linear operator. In the following \(y\in X\) is fixed and \(\alpha \in \mathbb {R}\).

1. Prove that there exists a constant \(C>0\) such that if \(\lvert \alpha \rvert <C\), then the non-linear equation \[ x+\alpha f(x)Tx=y, \tag {1} \] has a unique solution \(x\) in the ball \(B=\{x\in X\mid \lVert x-y\rVert \le 1\}\).
2. Suggest an iterative procedure for approximating the unique solution \(x\) to equation (1).

Let \(T:X\to X\) be a compact linear operator and \(S:X\to X\) be a bounded linear operator, defined on a Banach space \(X\).

1. Show that \(TS\) is a compact linear operator.
2. Show that \(ST\) is a compact linear operator.

Let \(V\) be a Banach space, and \(T:V\to V\) a bounded linear map. Assume that for every \(v\in V\) there exists a non-negative integer \(n\) such that \(T^n(v)=0\). Prove that there exists an integer \(n\) such that \(T^n(v)=0\) for all \(v\in V\).

Show that a real symmetric linear operator has a full set of real orthogonal eigenvectors (even if it has multiple eigenvalues). [Assume that the operator is a matrix acting in \(\mathbb {R}^n\).]

Let \(X\) be a normed space, and let \(X^*\) be the space of bounded linear functionals on \(X\). For every \(x\in X\), let \(x^*\) be the functional on \(X^*\) defined by \(x^*(y)=y(x)\) for all \(y\in X^*\). Prove that \(x\mapsto x^*\) is an isometry from \(X\) into \(X^{**}\).

Let \(V\) be a Banach space, and \(T:V\to V\) a bounded linear map. Assume that for every \(v\in V\) there exists a non-negative integer \(n\) such that \(T^n(v)=0\). Prove that there exists an integer \(n\) such that \(T^n(v)=0\) for all \(v\in V\).

Let \(T:X\to X\) be a bounded linear operator on a complex Banach space \(X\). Prove that the spectrum \(\sigma (T)\) lies in the complex plane disk: \[ \{\lambda \in \mathbb {C}:\lvert \lambda \rvert \le \lVert T\rVert \}. \]

Let \(T\) be a bounded linear operator on a complex Hilbert space \(H\). Show that the operator \(I+T^*T\) is injective.

Let \(T:X\to X\) be a compact linear operator on a normed space \(X\). Show that for every \(\lambda \ne 0\), the null space of \(T_\lambda =T-\lambda I\) is finite dimensional.

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 \(T:V\to U\) be a bounded linear map between two normed spaces. Let \(T^*:U^*\to V^*\) be defined by \(T^*(f)=f\circ T\) for all \(f\in U^*\) (the adjoint map). Prove that \(\lVert T^*\rVert =\lVert T\rVert \).

Let \(V\) be a Banach space, and \(T:V\to V\) a bounded linear map. Assume that for every \(v\in V\) there exists a non-negative integer \(n\) such that \(T^n(v)=0\). Prove that there exists an integer \(n\) such that \(T^n(v)=0\) for all \(v\in V\).

Let \(T:V\to W\) be a bounded linear map, where \(V\) and \(W\) are normed spaces. Let \(U\) be the kernel of \(T\). Prove that \(\lVert S\rVert =\lVert T\rVert \) where \(S:V/U\to W\) is such that \[ S(x+U)=T(x) \] for all \(x\in V\).

Let \(V\) be a Banach space, and \(T:V\to V\) a bounded linear map. Assume that for every \(v\in V\) there exists a non-negative integer \(n\) such that \(T^n(v)=0\). Prove that there exists an integer \(n\) such that \(T^n(v)=0\) for all \(v\in V\).

Let \(H\) be a Hilbert space. Show that an isometric linear operator \(T:H\to H\) satisfies \(T^*T=I\), where \(I\) is the identity operator on \(H\).

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 \[ E_\lambda x= \begin {cases} 0, & \lambda <0,\\ v_\lambda , & 0\le \lambda \le 1,\\ x, & \lambda >1, \end {cases} \qquad \text {where}\qquad v_\lambda (t)= \begin {cases} x(t), & 0\le t\le \lambda ,\\ 0, & \lambda <t\le 1. \end {cases} \]

Let \(X\) be a real Banach space. Assume \(f\in X^*\) has a closed nullspace \(\mathcal {N}(f)\). The goal of this problem is to show that \(f\) must be a bounded linear functional.

1. Let \(x_0\in X\) be such that \(f(x_0)=1\). Explain why there is an \(\epsilon >0\) for which the ball \[ B(x_0,\epsilon )\equiv \{x\in X\mid \lVert x-x_0\rVert <\epsilon \} \] satisfies \(B(x_0,\epsilon )\subset X-\mathcal {N}(f)\).
2. Prove that \(f(x)>0\) for all \(x\in B(x_0,\epsilon )\), where \(\epsilon \) is as in part (a). To prove this, you may assume for contradiction that there is some \(y\in B(x_0,\epsilon )\) with \(f(y)<0\).
3. Any \(x\in B(x_0,\epsilon )\) can be written as \(x=x_0+\epsilon u\) for some \(u\) with \(\lVert u\rVert <1\). Use the result of part (b) to show that \(\lvert f(u)\rvert <1/\epsilon \).
4. To conclude, give an upper bound for \(\lVert f\rVert \).

Let \(T:X\to X\) be a bounded linear operator on a complex Banach space \(X\). Prove that \(\sigma (T)\) lies in the complex plane disk: \[ \{\lambda \in \mathbb {C}\mid \lvert \lambda \rvert \le \lVert T\rVert \}. \]

Let \(X\) and \(Y\) be normed vector spaces. Show that a linear operator \(T:X\to Y\) is compact if and only if for every sequence \((x_n)\) with \(\lVert x_n\rVert \le 1\), the sequence \((Tx_n)\) has a convergent subsequence.

Let \(T\) be a bounded linear operator on a complex Hilbert space \(H\). Show that the operator \(I+T^*T\) is injective.

Let \(X\) be a Banach space, \(f:X\to \mathbb {R}\) be a bounded linear functional, \(y\in X\) fixed and \(\alpha \in \mathbb {R}\).

1. Prove that there exists a constant \(C>0\) such that if \(\lvert \alpha \rvert <C\), then the non-linear equation \[ x+\alpha f(x)x=y, \tag {2} \] has a unique solution \(x\) in the ball \(B=\{x\in X\mid \lVert x-y\rVert \le 1\}\).
2. Suggest an iterative procedure for approximating the unique solution \(x\) to equation (2).

Recall that if \(A\) is a linear operator on Hilbert space with bounded operator norm, then it has an adjoint \(A^*\) which is the unique operator so that for all \(v,w\), \[ \langle Av,w\rangle =\langle v,A^*w\rangle . \]

(a) Let \(P:\ell ^2(\mathbb N)\to \ell ^2(\mathbb N)\) by \(P\bar v=\bar w\) where \(w_j=0\) if \(j\) is not prime and \(w_j=v_j\) otherwise. What is \(P^*\)?

(b) Let \(P:\ell ^2(\mathbb N)\to \ell ^2(\mathbb N)\) by \(P\bar v=\bar w\) where \(w_j=v_{p_j}\), where \(p_j\) is the \(j\)th prime. What is \(P^*\)?

Let \(u,v\) be non-zero elements of a Hilbert space \(H\). Consider the linear operator \(T:H\to H\) defined for \(x\in H\) by \(Tx=u(x,v)\).

1. Briefly explain why the operator \(T\) is compact.
2. Find a condition on \(u\) and \(v\) that guarantees that the equation \[ (I-T)x=y \] admits a unique solution \(x\) for all \(y\in H\).

Let \(T:X\to X\) be a bounded linear operator defined on a Banach space \(X\). Denote by \(R_\lambda \) the resolvent operator associated with \(T\) for some \(\lambda \in \rho (T)\). The resolvent set of \(T\) is denoted by \(\rho (T)\).

1. Use Neumann series to show that for \(\lambda ,\mu \in \rho (T)\), \[ R_\lambda =\sum _{j=0}^{\infty }(\lambda -\mu )^j R_\mu ^{j+1}, \] where the series is absolutely convergent in the operator norm when \[ \lvert \lambda -\mu \rvert <\lVert R_\mu \rVert ^{-1}. \]
2. Deduce from part (a) whether the spectrum \(\sigma (T)\) is open, closed or neither.

Consider \(L^2(\mathbb Z):=H\). Recall two topologies on the set of bounded linear operators on \(H\): The norm topology which is metric topology given by \[ d(A,B)=\sup _{v\in H:\lVert v\rVert =1}\lVert Av-Bv\rVert . \] The strong operator topology which is given by \(A_i\to A_\infty \) if for every \(v\in H\) we have \[ \lim _{i\to \infty }\lVert A_iv-A_\infty v\rVert =0. \]

(a) Show that if \(A_i\) converges to \(A_\infty \) in the norm topology then it converges to \(A_\infty \) in the strong operator topology.

(b) Give an example of \(A_i\) which converges to \(A_\infty \) in the strong operator topology but not the norm topology.

(c) Let \(A_i\) be the operator defined by \(A_iv=w\) where \(w_j=v_j\) if \(j\ne i\) and \(w_i=iv_i\). Show \(A_1,\ldots \) is a sequence of operators that does not converge in either topology.

Let \(T:L^1[0,1]\to L^2[0,1]\) be a bounded linear operator. Let \(a\in L^\infty [0,1]\) and \(g\in L^1[0,1]\).

1. Prove that there exists a constant \(C>0\) such that if \(\lVert a\rVert _{L^\infty }<C\) then the equation \[ f+a(Tf)^2=g \tag {1} \] has a unique solution \(f\) in the set \[ B=\{u\in L^1[0,1]:\lVert u-g\rVert _{L^1}\le 1\}. \] Note: the constant \(C\) may depend on \(T\) and/or \(g\).
2. Suggest an iterative procedure for approximating the unique solution \(f\) to equation (1).

Let \(X\) be a separable Hilbert space and \((e_n)\) be a total orthonormal sequence of \(X\). Let \(T:X\to X\) be a bounded linear operator satisfying \[ \sum _{n=1}^{\infty }\lVert Te_n\rVert ^2<\infty . \tag {2} \] Define the operators \(T_k:X\to X\), \(k=1,2,\ldots \), that applied to an arbitrary \[ x=\sum _{n=1}^{\infty }x_ne_n\in X \qquad \text {give}\qquad T_kx=\sum _{n=1}^k x_nTe_n. \]

1. Are the operators \(T_k\) compact? Explain why or why not.
2. Using equation (2), show that the sequence \((T_k)\) converges uniformly to \(T\), i.e. show that \(\lVert T-T_k\rVert \to 0\) as \(k\to \infty \).
3. Is \(T\) compact? Explain why or why not.

Let \(K:L^2[0,1]\to L^2[0,1]\) be the linear operator defined by \[ (Kx)(t)=\int _0^1 k(t,\tau )x(\tau )\,d\tau \] where \(k:[0,1]\times [0,1]\to \mathbb {R}\) is continuous.

1. Show that \(K\) is bounded and give an upper bound for \(\lVert K\rVert \) (in terms of \(k\)).
2. Show that there is a constant \(C\) such that if \(\lvert \mu \rvert <C\), the operator \(I+\mu K\) is invertible.
3. Assuming \(\lvert \mu \rvert <C\), where \(C\) is as in part (b), express \((I+\mu K)^{-1}\) as a series. Specify whether the series converges and in what sense.

Let \(V\) and \(W\) be Banach spaces, and \(L(V,W)\) the space of bounded linear maps from \(V\) to \(W\). Suppose that \(T_n\in L(V,W)\), \(n=1,2,\ldots \), and that \(\lim T_nx\) exists for each \(x\in V\). Show that \(T:V\to W\) defined by \(Tx=\lim T_nx\) is in \(L(V,W)\).

Let \(A\) be an \(n\times n\) real matrix. Consider the linear initial value problem \[ \begin {cases} \dfrac {dx}{dt}=Ax,\\ x(0)=x_0. \end {cases} \] Give an algebraic condition on the matrix \(A\) which holds if and only if all solutions of the initial value problem are bounded for all \(t\in \mathbb {R}\). Prove the equivalence.

Suppose that \(E\) is a bounded linear operator on a Hilbert space \(H\), with operator norm \(\lVert E\rVert =r<1\). Show that the operator \(I-E\), where \(I\) is the identity, is bounded and invertible. Give an estimate for \(\lVert (I-E)^{-1}\rVert \), and a convergent series representation for \((I-E)^{-1}\).

Consider linear operator \[ H=-\frac {d^2}{dx^2}, \] acting on \(\psi (x)\in L^2[0,\pi ]\) with periodic boundary conditions \(\psi (0)=\psi (\pi )=0\). Find the eigenvalues and eigenfunctions of \(H\). Show that the eigenvectors of \(H\) corresponding to distinct eigenvalues are orthogonal. Are the eigenvectors of \(H\) complete, and if so, in what sense?

(b) Solve the equation \[ H\psi (x)=x^2, \] representing the solution as an expansion in eigenfunctions of \(H\).

Let \((X,\mu )\) be a finite measure space, and let \(f\in L^\infty (X)\). For \(g\in L^1(X)\), define \[ Tg=\int fg\,d\mu \] Show that \(T\) is a bounded linear functional on \(L^1(X)\), and that \(\lVert T\rVert =\lVert f\rVert _\infty \).

Let \(X\) and \(Y\) be Banach spaces, and suppose that \(T:X\to Y\) is a compact surjective linear map. Show that \(Y\) is finite dimensional. (Recall that a linear map is compact if the image of any bounded set has compact closure.)

1. Define what is meant by a precompact (or sequentially compact) set and what is meant by a compact linear operator, assuming linear operators and bounded sets have been already been defined.
2. If \(K\) is a compact linear operator and if \(\{\phi _n\}\) is an infinite set of orthonormal functions then (using Bessel’s inequality, if needed) show that \[ \lim _{n\to \infty }K\phi _n=0. \]

Let \(T:C_0(\mathbb R)\to C_0(\mathbb R)\) be the translation operator defined by \((T.f)(x)=f(x+1)\). Show that there is no non-zero bounded linear functional \(\Phi :C_0(\mathbb R)\to \mathbb C\) invariant under \(T\), i.e. such that \(\Phi \) satisfies \[ \Phi (f)=\Phi (Tf). \]

Consider the scalar linear equation \[ a\frac {\partial u}{\partial x}+b\frac {\partial u}{\partial y}=\alpha u \] with \(a,b,\alpha \) differentiable functions of \(x,y\). Suppose that \(u\) is prescribed on some arc \(\Gamma \) (Cauchy data).

1. By introducing a test function \(\psi \) that vanishes on an arbitrary curve \(\gamma \) and applying Green’s theorem to the domain \(D\) bounded by \(\partial D=\Gamma \cup \gamma \), derive the integral equation for a weak solution to the Cauchy problem.
2. Extend the analysis to the case where \(u\) is discontinuous across an open curve \(C_0\) within the domain \(D\). Hence show that \(C_0\) is a characteristic projection.
3. Write down the generalization of part (b) to the case of the quaslinear equation \[ \frac {\partial P}{\partial x}+\frac {\partial Q}{\partial y}=c \] where \(P,Q,c\) are differentiable functions of \(x,y,u\), and hence derive the Rankine–Hugonoit condition.

Prove that if a linear operator \(K\) can be approximated (in the operator norm) by a sequence of compact operators \(K_n\), with, \[ \lim _{n\to \infty }\lVert K_n-K\rVert =0, \] then \(K\) is a compact operator.

Let \(H\) be a Hilbert space, and let \(T:H\to H\) be a self adjoint linear operator, with finite dimensional range \(V=T(H)\). Prove that \(T\) is a compact operator, and that \(V=K^\perp \), where \(K\) is the kernel of \(T\).

Assume that \(\mathbb R\) has the Lebesgue measure. Let \(f\in L^\infty (\mathbb R)\). For \(g\in L^2(\mathbb R)\) define \[ M(g)=fg \] Prove that \(M:L^2(\mathbb R)\to L^2(\mathbb R)\) is a bounded linear operator, and that \(\lVert M\rVert =\lVert f\rVert _\infty \).

1. 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 \(H^1(G)\).
2. State the Lax-Milgram theorem for subspaces \(V\subset H^1(G)\).

3. Show that the Lax-Milgram theorem may be applied to the form \[ a(u,v)=\sum _{i=1}^{N}\int _G \partial _i u\,\partial _i\overline {v} +\int _G u\overline {v} \] and \(f\) as above.

   Assume that \(G\) is a bounded open set with “very smooth” boundary and let \(u\) be the Lax-Milgram solution just determined. What partial differential equation (in the sense of distributions) will be solved by \(u\) and what boundary conditions does \(u\) satisfy (please give an argument to prove the latter)?

1. Let \(H\) be a Hilbert space and let \[ A:D(A)\to H \] be a linear operator with domain \(D(A)\). State the Hille-Yosida-Phillips theorem giving necessary and sufficient conditions for the unique solvability of the problem \[ \begin {aligned} u'+Au&=0\\ u(0)&=u_0\in D(A). \end {aligned} \] Be sure to define pertinent terms.
2. Apply the above result to the initial boundary value problem \[ \begin {aligned} u_t-u_x&=0,\quad t>0,\quad x\in \mathbb {R}\\ u(0,x)&=g(x),\quad x\in \mathbb {R}. \end {aligned} \] Also solve this problem explicitly.

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.

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 \[ (Ku)(y)=\int _0^1 \min (x,y)u(x)\,dx. \]

The space \(C[0,1]\) of continuous functions on \([0,1]\) is a subspace of all the \(L^p\) spaces, \(L^p[0,1]\) for \(1\le p\le \infty \). Define a linear operator \(T:C[0,1]\to \mathbb R\) by \(Tf=f(1/2)\). Show that \(T\) is a bounded operator for the \(L^p\)-norm if and only if \(p=\infty \). Show that \(T\) extends to a bounded operator on all of \(L^\infty [0,1]\).

In what follows, \(D_R(z_0)\) and \(\overline {D_R}(z_0)\) denote the open and closed discs, respectively, centered at \(z_0\), with radius \(R\).

Let \(K:H\to H\) be a compact linear operator on the infinite-dimensional Hilbert space \(H\).

1. Prove that \(K^{-1}\), if it exists, is not bounded.
2. Given \(T>0\), let \(H=L^2(0,T)\). For \(u\in H\), define \[ (Ku)(t)=\int _0^T e^{-st}u(s)\,ds,\qquad 0\le t\le T. \] Prove that \(K:H\to H\) is a compact operator. What are the consequences of this fact for the utility of the Laplace transform as a numerical method?

Let \(f\) be a bounded linear functional on a linear subspace \(M\) of a Hilbert space \(H\).

(i) Prove that \(f\) has a unique extension to a bounded linear functional \(F\) on \(H\) so that the norm of \(f\) coincides with the norm of \(F\).

(ii) Prove that the extension \(F\) vanishes on \(M^\perp \).

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.

Let \(X=[0,1]\), and \(C(X)\) the space of complex valued continuous functions on \(X\), equipped with the sup norm: \(\|f\|_\infty =\sup _{x\in X}|f(x)|\). Let \(f_n\) be a sequence in \(C(X)\). Prove that the following statements are equivalent:

(a) For each bounded linear functional \(\lambda \) on \(C(X)\), \(\lambda (f_n)\to 0\);

(b) \(f_n(x)\to 0\) for each \(x\in X\), and \(\sup _n\|f_n\|_\infty \) is finite.

Let \(\lambda \) be a bounded linear functional on a linear subspace \(V\) of a Hilbert space \(H\). Show that there exists a unique bounded linear functional \(\Lambda \) on \(H\) such that: (a) \(\|\Lambda \|=\|\lambda \|\), and (b) \(\Lambda (x)=\lambda (x)\) for \(x\in V\).

Prove that if \(X\) is a Banach space with a closed unit ball which is compact, then \(X\) must be finite dimensional. Hint: consider the family of sets of the form \(\{x\in X:\|x\|=1,\text { and }f(x)=0\}\) where \(f\) is a bounded nonzero linear functional on \(X\).

1. Let \(H\) be a Hilbert space and let \[ A:D(A)\to H \] be a linear operator with domain \(D(A)\). State the Hille-Yosida-Phillips theorem giving necessary and sufficient conditions for the unique solvability of the problem \[ \begin {aligned} u'+Au&=0,\\ u(0)&=u_0\in D(A). \end {aligned} \] Be sure to define pertinent terms.

2. Apply the above result to the initial boundary value problem \[ \begin {aligned} u_t-u_x&=0,\qquad t>0,\ x\in \mathbb {R},\\ u(0,x)&=g(x),\qquad x\in \mathbb {R}. \end {aligned} \] Solve this problem explicitly.

   Also state a result about the initial value problem \[ \begin {aligned} u_t-u_x&=f(t,x),\qquad t>0,\ x\in \mathbb {R},\\ u(0,x)&=g(x),\qquad x\in \mathbb {R}. \end {aligned} \]

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 \[ (Ku)(y)=\int _0^1 \min (x,y)u(x)\,dx. \]

Let \(X\) be a Banach space and let \(T\) be a function which assigns to each real number \(s\in \mathbb R\) a bounded linear operator \(T(s):X\to X\). The derivative \(T'(s)\) of \(T\) at \(s\) is the operator defined by \[ T'(s)x=\lim _{h\to 0}h^{-1}(T(s+h)-T(s))x \] where the domain of \(T'(s)\) is the set of all \(x\in X\) for which this limit exists. Prove that if the domain of \(T'(s)\) is all of \(X\) for a given \(s\), then \(T'(s)\) is a bounded linear operator on \(X\).

Let \(X=[0,1]\), and \(C(X)\) the space of complex valued continuous functions on \(X\), equipped with the sup norm: \(\|f\|_\infty =\sup _{x\in X}|f(x)|\). Let \(f_n\) be a sequence in \(C(X)\). Prove that the following statements are equivalent:

(a) For each bounded linear functional \(\lambda \) on \(C(X)\), \(\lambda (f_n)\to 0\);

(b) \(f_n(x)\to 0\) for each \(x\in X\), and \(\sup _n\|f_n\|_\infty \) is finite.

Let \(X\) be a compact Hausdorff space. Let \(C(X)\) be the space of complex valued continuous functions on \(X\). We assume that \(C(X)\) is equipped with a norm \(\|\cdot \|\), which turns it into a Banach space. Assume moreover, that, for each \(x\in X\) the linear functional \(\lambda _x\), defined by: \(\lambda _x(f)=f(x)\) is bounded on this Banach space. Show that there exist positive constants \(A\) and \(B\), such that for every \(f\in C(X)\) \[ A\|f\|_\infty \le \|f\|\le B\|f\|_\infty . \] Here \(\|f\|_\infty \) denotes, as usual, the sup norm.

Let \(H\) be a Hilbert space and let \[ A:D(A)\to H \] be a linear operator with domain \(D(A)\). State the Hille-Yosida-Phillips theorem giving necessary and sufficient conditions for the unique solvability of the problem \[ \begin {aligned} u'+Au&=0,\\ u(0)&=u_0\in D(A). \end {aligned} \] Be sure to define pertinent terms.

A bounded linear transformation \(A\) from a Banach space \(X\) to a Banach space \(Y\) is said to be ”compact” if the image under \(A\) of the unit ball in \(X\) has compact closure in \(Y\). Prove that if a compact linear transformation is also surjective, then \(Y\) is finite dimensional (you may assume that a locally compact Banach space must be finite dimensional).

Let \(T\) be a bounded linear functional on a (linear) subspace \(M\) of a Hilbert space \(H\). Prove that \(T\) has a unique norm preserving extension to a bounded linear functional on \(H\), and that this extension vanishes on \(M^\perp \).

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 \(L^*\).

Let \(\ell ^\infty \) be the normed space consisting of all complex valued, bounded sequences \[ \mathbf a=(a_1,a_2,\cdots a_n\cdots ) \] with the norm \(\lVert \mathbf a\rVert =\sup \{|a_n|:n\in \mathbb N\}\). Let \(W\subset \ell ^\infty \) be the subspace consisting of all \(\mathbf a\in \ell ^\infty \) such that \(\{n\in \mathbb N:a_n\ne 0\}\) is a finite set.

Show that there exists a nontrivial bounded linear functional on \(\ell ^\infty \) which vanishes on \(W\).

Let \(H\) be a Hilbert space and let \[ A:D(A)\to H \] be a linear operator with domain \(D(A)\). State the Hille-Yosida-Phillips theorem giving necessary and sufficient conditions for the unique solvability of the problem \[ \begin {aligned} u'+Au&=0\\ u(0)&=u_0\in D(A). \end {aligned} \] Be sure to define pertinent terms.

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

Let \(B\) be a Banach space and \(A\subset B\) a closed subspace. If \(x\in B\) but \(x\notin A\), prove that there is a bounded linear functional \(f\) on \(B\) such that \(f(x)=1\), \(f(y)=0\) for every \(y\in A\) and \(\lVert f\rVert =(\inf \{\lVert x+y\rVert :y\in A\})^{-1}\).

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 \(x\in \mathcal {H}\).)

Let \(A\), \(B\), and \(C\) be Banach spaces and let \(f:A\to C\) and \(g:B\to C\) be bounded linear maps. Suppose that \(g\) is one to one and \(f(A)\subset g(B)\). Prove that there is a bounded linear map \(h:A\to B\) such that \(b=h(a)\) if and only if \(f(a)=g(b)\).

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 \[ \mathcal {H}\oplus \mathcal {H}=V(\overline {G(T)})\oplus G(T^*). \]

Let \(X=[a,b]\subset \mathbb R\) be a closed interval. Assume \(a<b\). Regard \(X\) as a measure space with respect to the usual Lebesgue measure.

(a) Show directly that the space \(C(X)\) of continuous functions on \(X\) is not dense in \(L^\infty (X)\).

(b) Using (a), if necessary, show that there exists a bounded linear functional on \(L^\infty (X)\) which vanishes on \(C(X)\).

State necessary and sufficient conditions in order that a linear operator be the generator of a strongly continuous contraction semigroup on \(H\).

Let \(f\in L^\infty (\mathbb R)\) define a bounded linear operator \(M_f\) from \(L^2(\mathbb R)\) to \(L^2(\mathbb R)\) by \(M_f(g)=fg\).

(a) Prove that \(\lVert M_f\rVert =\lVert f\rVert _\infty \).

(b) Prove that if \(M_f\) is one to one and onto then there exists \(\epsilon >0\) and a set \(E\) of measure zero such that \(|f(x)|\ge \epsilon \) for \(x\in \mathbb R-E\).