Chapter 13

Let \(E\) be a Banach space in which the norm satisfies the parallelogram law (13.2). Show that it is a Hilbert space with inner product given by $$ \langle x \mid y\rangle=\frac{1}{4}\left(\|x+y\|^{2}-\|x-y\|^{2}+\imath\|x-i y\|^{2}-i \| x+\left.1 y\right|^{2}\right) $$

On the vector space \(\mathcal{F}^{\prime}[a, b]\) of complex continuous differentiable functions on the, interval \([a, b]\), set $$ \langle f| g)=\int_{a}^{b} \overline{f^{\prime}(x)} g^{\prime}(x) \mathrm{dr} \text { where } f^{\prime}=\frac{\mathrm{d} f}{\mathrm{~d} x}, \quad g^{\prime}=\frac{\mathrm{d} g}{\mathrm{~d} x} $$ Show that this is not an inner product, but becomes one if restricted to the space of functions \(f \in\) \(F^{\prime}[a, b]\) having \(f(c)=0\) for seme fixed \(a \leq c \leq b\). Is it a Hilbert space? Give a similar analysis for the case \(a=-\infty, b=\infty\), and restricting functions to those of compact support.

In the space \(L^{2}([0,1])\) which of the following sequences of functions (i) is a Cauchy sequence, (ii) converges to 0 , (iii) converges everywhere to 0, (iv) converges almost everywhere to 0 , and (v) converges almost nowhere to \(0 ?\). (a) \(f_{n}(x)=\sin ^{n}(x), n=1,2, \ldots\) (b) \(f_{n}(x)= \begin{cases}0 & \text { for } x<1-\frac{1}{n}, \\ n x+1-n & \text { for } 1-\frac{1}{n} \leq x \leq 1 .\end{cases}\) (c) \(f_{n}(x)=\sin ^{n}(n x)\) (d) \(f_{n}(x)=\chi_{L_{s}}(x)\), the characteristic function of the set \(U_{n}=\left[\frac{k}{2^{m}}, \frac{k+1}{2^{m}}\right]\) where \(n=2^{m}+k, m=0,1, \ldots\) and \(k=0 . \ldots .2^{m}-1\)

Show that a vector subspace is a closed subset of \(\mathcal{H}\) with respect to the norm topology iff the limit of every sequence of vectors in \(V\) belongs to \(V\).

Let \(\ell_{0}\) be the subset of \(\ell^{2}\) consisting of sequences with only finutely many terms different from zero. Show that \(\ell_{0}\) is a vector subspace of \(\ell^{2}\), but that it is not closed. What is its closure \(\overline{\ell_{0}} ?\)

In the Hulbert space \(L^{2}([-1,1])\) let \(\left.\mid f_{n}(x)\right\\}\) be the sequence of functions \(1, x, x^{2}, \ldots, f_{n}(x)=x^{n} \ldots .\) (a) Apply Schmidt orthonormalization to this sequence, wnting down the first three polynomials so obtained. (b) The \(n\)th Legendre polynomial \(P_{n}(x)\) is defined as $$ P_{n}(x)=\frac{1}{2^{n} n !} \frac{d^{n}}{\mathrm{dx}^{n}}\left(x^{2}-1\right)^{n} $$ Prove that $$ \int_{-1}^{1} P_{m}(\mathrm{r}) P_{n}(\mathrm{x}) \mathrm{d} \mathrm{x}=\frac{2}{2 n+1} \delta_{m \mathrm{~m}} $$ (c) Show that the \(n\)th member of the o.n. sequence obtained in (a) is \(\sqrt{n+\frac{1}{2}} P_{n}(x)\).

If \(S\) is any subset of \(\mathcal{H}\), and \(V\) the closed subspace generated by \(S, V=\overline{L(S)}\), show that \(S^{1}=\left\\{u \in \mathcal{H} \mid\\{u|x\rangle=0\right.\) for all \(x \in S\\}=V^{1}\)

Which of the following is a vector subspace of \(\ell^{2}\), and which are closed? In each case find the space of vectors orthogonal to the set. (a) \(V_{N}=\left\\{\left(x_{1}, x_{2} \ldots\right) \in \ell^{2} \mid x_{1}=0\right.\) for \(\left.i>N\right]\) (b) \(V=\bigcup_{N=1}^{\infty} V_{N}=\left|\left(x_{1}, x_{2}, \ldots\right) \in \ell^{2}\right| x_{t}=0\) for \(i>\) some \(\left.N\right]\). (c) \(U=\left|\left(x_{1}, x_{2}, \ldots\right) \in \ell^{2}\right| x_{1}=0\) for \(\left.t=2 n\right\\} .\) (d) \(W=\left\\{\left(x_{1}, x_{2}, \ldots\right) \in \ell^{2} \mid x_{1}=0\right.\) for some \(\left.i\right\\}\)

If \(A: \mathcal{H} \rightarrow \mathcal{H}\) is an operator such that \(A u \perp u\) for all \(u \in \mathcal{H}\), show that \(A=0\).

The norm \(\|\phi\|\) of a bounded linear operator \(\phi: \mathcal{H} \rightarrow \mathrm{C}\) is defined as the greatest lower bound of all \(M\) such that \(|\phi(u)| \leq M\|u\|\) for all \(u \in \mathcal{H}\). If \(\phi(u)=(v \mid u)\) show that \(\|\phi\|=\|v\|\). Hence show that ahe bounded lanear functional norm satisfies the parallelogram law $$ \|\phi+\vartheta\|^{2}+\|\phi-\psi\|^{2}=2\|\phi\|^{2}+2\|\psi\|^{2} $$

For bounded linear operators \(A, B\) on a normed vector space \(V\) show that $$ \|\lambda A\|=|\lambda|\|A\|, \quad|A+B\|\leq\| A|+\|B\|, \quad \mid A B\|\leq\| A\|\| B \| $$ Hence show that \(|A|\) is a genuine norm on the set of bounded hnear operators on \(V\).

Let \(A\) be a bounded openitor on a Hilbert space \(\mathcal{H}\) with a one- damensional rangc. (a) Show that there exist vectors \(u, v\) such that \(A x=\langle v \mid x\rangle u\) for all \(x \in \mathcal{H}\). (b) Show that \(A^{2}=\lambda A\) for some scalar \(\lambda\), and that \(\|A\|=\|u\|\|v\|\). (c) Prove that \(A\) is hermitian, \(A^{*}=A\), If and only if there exists a real number \(a\) such that \(v=a u\).

For every bounded operator \(A\) on a Hilbert space \(\mathcal{H}\) show that the exponential operator $$ \mathrm{e}^{A}=\sum_{n=0}^{\infty} \frac{A^{n}}{n !} $$ is well-defined and bounded on \(\mathcal{H}\). Show that (a) \(e^{0}=1\) (b) For all posituve integers \(n,\left(c^{A}\right)^{n}=e^{n A}\). (c) \(\mathrm{e}^{A}\) is invertuble for all bounded operators \(A\) (even if \(A\) is not mivertible) and \(e^{-A}=\left(e^{4}\right)^{-1} .\) (d) If \(A\) and \(B\) are commuting operators then \(e^{A+B}=\mathrm{e}^{A} \mathrm{e}^{\theta}\) (c) If \(A\) is hermitian then \(e^{i A}\) is unitary.

Show that the sum of two projection operators \(P_{u}+P_{M}\) is a projection operator iff \(P_{M} P_{N}=0\). Show that this condition is equavalent to \(M \perp N\).

Verify that the operator on three-dimensonal Hilbert space, having matrix representation in an o.n. basis $$ \left(\begin{array}{ccc} \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & 1 & 0 \\ -\frac{1}{2} & 0 & \frac{1}{2} \end{array}\right) $$ is a projection operator, and find a basis of the subspace it projects onto.

Let \(\omega=c^{2 \pi t / 3} .\) Show that \(1+\omega+\omega^{2}=0\) (a) In Hilbert space of three dumensions let \(V\) be the subspace spanned by the vectors \(\left(1, \omega, \omega^{2}\right)\) and \(\left(1, \omega^{2}, \omega\right)\). Find the vector \(u_{0}\) in this subspece that is closess to the vector \(u=(1,-1,1)\). (b) Verify that \(u-u_{0}\) is orthogonal to \(V\). (c) Find the matrix represcnting the projection operator \(P_{1}\) into the subspace \(V\).

An operator \(A\) is called nermal if it is bounded and commutes with its adjoint. \(A^{*} A=A A^{*} .\) Show that the operator $$ A \psi(x)=c \psi(x)+l \int_{a}^{t} K(x, y) \psi(y) \mathrm{d} y $$ on \(L^{2}([a, b])\), where \(c\) is a real number and \(K(x, y)=\overline{K(y, x)}\), is normal. (a) Show that an operator \(A\) is normal if and only if \(\|A u\|=\left\|A^{*} u\right\|\) for all vectors \(u \in \mathcal{H}\). (b) Show that if \(A\) and \(B\) are commuting normal operators, \(A B\) and \(A+\lambda B\) are normal for all \(\lambda \in \mathbb{C}\)

Show that a non-zero vector \(u\) is an cigenvector of an operator \(A\) if and only if \(|\langle u \mid A u\rangle|=\| A u|| u \mid .\)

Show that every complex number \(\lambda\) in the spectrum of a unitary operator has \(|\lambda|=1\).

For any pair of hermitian operators \(A\) and \(B\) on a Hilbert space \(\mathcal{H}\), define \(A \leq B\) iff \((u \mid A u) \leq(u \mid B u)\) for all \(u \in \mathcal{H}\). Show that this is a partial order on the set of hermatian operators pay particular attention to the symmetry property, \(A \leq B\) and \(B \leq A\) umplies \(A=B\). (a) For multiplication operators on \(L^{2}(X)\) show that \(A_{0} \leq A_{B}\). Iff \(\alpha(x) \leq \beta(x)\) a.e. on \(X\). (b) For projection operators show that the definition given here reduces to that given in the text, \(P_{N} \leq P_{N}\) iff \(M \subseteq N\)

Show that if \(\left(A, D_{A}\right)\) and \(\left(B, D_{B}\right)\) arc operators on dense domains in \(H\) then \(B^{*} A^{*} \subseteq\) \((A B)^{\circ}\)

For unbounded operators, show that \(A^{*}+B^{*} \subseteq(A+B)^{\circ}\)

If \(\left(A, D_{A}\right)\) is a densely defined openator and \(D_{A}\) is dense in \(\mathcal{H}_{1}\) show that \(A \subseteq A^{* *}\).

If \(A\) is a symmetric operator, show that \(A^{*}\) is symmetric if and only if it is self-edjoint, \(A^{*}=A^{* *}\)

If \(A_{1}, A_{2}, \ldots . A_{n}\) are operators on a dense domain such that $$ \sum_{i=1}^{n} A_{1}^{*} A_{1}=0 $$ show that \(A_{1}=A_{2}=\cdots=A_{n}=0 .\)

If \(A\) is a sclf-adjoint operator show that $$ \|(A+t) u\|^{2}=\|A u\|^{2}+I u \|^{2} $$ and that the operator \(A+1 I\) is invertible, Show that the operator \(U=\left(A-{ }_{1} I\right)(A+i I)^{-1}\) is unitary (called the Cayley transform of \(A\) ).