Hilbert Spaces

# Hilbert Spaces

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B1.10

commentSuppose that $\left(e_{n}\right)$ and $\left(f_{m}\right)$ are orthonormal bases of a Hilbert space $H$ and that $T \in L(H)$.

(a) Show that $\sum_{n=1}^{\infty}\left\|T\left(e_{n}\right)\right\|^{2}=\sum_{m=1}^{\infty}\left\|T^{*}\left(f_{m}\right)\right\|^{2}$.

(b) Show that $\sum_{n=1}^{\infty}\left\|T\left(e_{n}\right)\right\|^{2}=\sum_{m=1}^{\infty}\left\|T\left(f_{m}\right)\right\|^{2}$.

$T \in L(H)$ is a Hilbert-Schmidt operator if $\sum_{n=1}^{\infty}\left\|T\left(e_{n}\right)\right\|^{2}<\infty$ for some (and hence every) orthonormal basis $\left(e_{n}\right)$.

(c) Show that the set HS of Hilbert-Schmidt operators forms a linear subspace of $L(H)$, and that $\langle T, S\rangle=\sum_{n=1}^{\infty}\left\langle T\left(e_{n}\right), S\left(e_{n}\right)\right\rangle$ is an inner product on $H S$; show that this inner product does not depend on the choice of the orthonormal basis $\left(e_{n}\right)$.

(d) Let $\|T\|_{H S}$ be the corresponding norm. Show that $\|T\| \leqslant\|T\|_{H S}$, and show that a Hilbert-Schmidt operator is compact.

B3.8

commentLet $H$ be a Hilbert space. An operator $T$ in $L(H)$ is normal if $T T^{*}=T^{*} T$. Suppose that $T$ is normal and that $\sigma(T) \subseteq \mathbb{R}$. Let $U=(T+i I)(T-i I)^{-1}$.

(a) Suppose that $A$ is invertible and $A T=T A$. Show that $A^{-1} T=T A^{-1}$.

(b) Show that $U$ is normal, and that $\sigma(U) \subseteq\{\lambda:|\lambda|=1\}$.

(c) Show that $U^{-1}$ is normal.

(d) Show that $U$ is unitary.

(e) Show that $T$ is Hermitian.

[You may use the fact that, if $S$ is normal, the spectral radius of $S$ is equal to $\|S\| .$ ]

B4.7

commentSuppose that $T$ is a bounded linear operator on an infinite-dimensional Hilbert space $H$, and that $\langle T(x), x\rangle$ is real and non-negative for each $x \in H$.

(a) Show that $T$ is Hermitian.

(b) Let $w(T)=\sup \{\langle T(x), x\rangle:\|x\|=1\}$. Show that

$\|T(x)\|^{2} \leqslant w(T)\langle T(x), x\rangle \quad \text { for each } x \in H$

(c) Show that $\|T\|$ is an approximate eigenvalue for $T$.

Suppose in addition that $T$ is compact and injective.

(d) Show that $\|T\|$ is an eigenvalue for $T$, with finite-dimensional eigenspace.

Explain how this result can be used to diagonalise $T$.

B1.10

commentLet $H$ be a Hilbert space and let $T \in \mathcal{B}(H)$.

(a) Define what it means for $T$ to be (i) invertible, and (ii) bounded below. Prove that $T$ is invertible if and only if both $T$ and $T^{*}$ are bounded below.

(b) Define what it means for $T$ to be normal. Prove that $T$ is normal if and only if $\|T x\|=\left\|T^{*} x\right\|$ for all $x \in H$. Deduce that, if $T$ is normal, then every point of Sp $T$ is an approximate eigenvalue of $T$.

(c) Let $S \in \mathcal{B}(H)$ be a self-adjoint operator, and let $\left(x_{n}\right)$ be a sequence in $H$ such that $\left\|x_{n}\right\|=1$ for all $n$ and $\left\|S x_{n}\right\| \rightarrow\|S\|$ as $n \rightarrow \infty$. Show, by direct calculation, that

$\left\|\left(S^{2}-\|S\|^{2}\right) x_{n}\right\|^{2} \rightarrow 0 \quad \text { as } n \rightarrow \infty$

and deduce that at least one of $\pm\|S\|$ is an approximate eigenvalue of $S$.

(d) Deduce that, with $S$ as in (c),

$r(S)=\|S\|=\sup \{|\langle S x, x\rangle|: x \in H,\|x\|=1\}$

B3.8

commentLet $\mathcal{H}$ be the space of all functions on the real line $\mathbb{R}$ of the form $p(x) e^{-x^{2} / 2}$, where $p$ is a polynomial with complex coefficients. Make $\mathcal{H}$ into an inner-product space, in the usual way, by defining the inner product to be

$\langle f, g\rangle=\int_{-\infty}^{\infty} f(t) \overline{g(t)} d t, \quad f, g \in \mathcal{H}$

You should assume, without proof, that this equation does define an inner product on $\mathcal{H}$. Define the norm by $\|f\|_{2}=\langle f, f\rangle^{1 / 2}$ for $f \in \mathcal{H}$. Now define a sequence of functions $\left(F_{n}\right)_{n \geqslant 0}$ on $\mathbb{R}$ by

$F_{n}(x)=(-1)^{n} e^{x^{2} / 2} \frac{d^{n}}{d x^{n}} e^{-x^{2}}$

Prove that $\left(F_{n}\right)$ is an orthogonal sequence in $\mathcal{H}$ and that it spans $\mathcal{H}$.

For every $f \in \mathcal{H}$ define the Fourier transform $\widehat{f}$ of $f$ by

$\widehat{f}(t)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(x) e^{-i t x} d x, \quad t \in \mathbb{R}$

Show that

(a) $\widehat{F}_{n}=(-i)^{n} F_{n}$ for $n=0,1,2, \ldots$;

(b) for all $f \in \mathcal{H}$ and $x \in \mathbb{R}$,

$\widehat{\widehat{f}}(x)=f(-x)$

(c) $\|\widehat{f}\|_{2}=\|f\|_{2}$ for all $f \in \mathcal{H}$.

B4.7

commentLet $H$ be a Hilbert space and let $T \in \mathcal{B}(H)$.

(a) Show that if $\|I-T\|<1$ then $T$ is invertible.

(b) Prove that if $T$ is invertible and if $S \in \mathcal{B}(H)$ satisfies $\|S-T\|<\left\|T^{-1}\right\|^{-1}$, then $S$ is invertible.

(c) Define what it means for $T$ to be compact. Prove that the set of compact operators on $H$ is a closed subset of $\mathcal{B}(H)$.

(d) Prove that $T$ is compact if and only if there is a sequence $\left(F_{n}\right)$ in $\mathcal{B}(H)$, where each operator $F_{n}$ has finite rank, such that $\left\|F_{n}-T\right\| \rightarrow 0$ as $n \rightarrow \infty$.

(e) Suppose that $T=A+K$, where $A$ is invertible and $K$ is compact. Prove that then, also, $T=B+F$, where $B$ is invertible and $F$ has finite rank.

B1.10

commentLet $H$ be a Hilbert space and let $T \in \mathcal{B}(H)$. Define what it means for $T$ to be bounded below. Prove that, if $L T=I$ for some $L \in \mathcal{B}(H)$, then $T$ is bounded below.

Prove that an operator $T \in \mathcal{B}(H)$ is invertible if and only if both $T$ and $T^{*}$ are bounded below.

Let $H$ be the sequence space $\ell^{2}$. Define the operators $S, R$ on $H$ by setting

$S(\xi)=\left(0, \xi_{1}, \xi_{2}, \xi_{3}, \ldots\right), \quad R(\xi)=\left(\xi_{2}, \xi_{3}, \xi_{4}, \ldots\right),$

for all $\xi=\left(\xi_{1}, \xi_{2}, \xi_{3}, \ldots\right) \in \ell^{2}$. Check that $R S=I$ but $S R \neq I$. Let $D=\{\lambda \in \mathbb{C}:|\lambda|<$ $1\}$. For each $\lambda \in D$, explain why $I-\lambda R$ is invertible, and define

$R(\lambda)=(I-\lambda R)^{-1} R$

Show that, for all $\lambda \in D$, we have $R(\lambda)(S-\lambda I)=I$, but $(S-\lambda I) R(\lambda) \neq I$. Deduce that, for all $\lambda \in D$, the operator $S-\lambda I$ is bounded below, but is not invertible. Deduce also that $\operatorname{Sp} S=\{\lambda \in \mathbb{C}:|\lambda| \leqslant 1\}$.

Let $\lambda \in \mathbb{C}$ with $|\lambda|=1$, and for $n=1,2, \ldots$, define the element $x_{n}$ of $\ell^{2}$ by

$x_{n}=n^{-1 / 2}\left(\lambda^{-1}, \lambda^{-2}, \ldots, \lambda^{-n}, 0,0, \ldots\right) .$

Prove that $\left\|x_{n}\right\|=1$ but that $(S-\lambda I) x_{n} \rightarrow 0$ as $n \rightarrow \infty$. Deduce that, for $|\lambda|=1, S-\lambda I$ is not bounded below.

B3.8

commentLet $H$ be an infinite-dimensional, separable Hilbert space. Let $T$ be a compact linear operator on $H$, and let $\lambda$ be a non-zero, approximate eigenvalue of $T$. Prove that $\lambda$ is an eigenvalue, and that the corresponding eigenspace $E_{\lambda}(T)=\{x \in H: T x=\lambda x\}$ is finite-dimensional.

Let $S$ be a compact, self-adjoint operator on $H$. Prove that there is an orthonormal basis $\left(e_{n}\right)_{n \geqslant 0}$ of $H$, and a sequence $\left(\lambda_{n}\right)_{n \geqslant 0}$ in $\mathbb{C}$, such that (i) $S e_{n}=\lambda_{n} e_{n}(n \geqslant 0)$ and (ii) $\lambda_{n} \rightarrow 0$ as $n \rightarrow \infty$.

Now let $S$ be compact, self-adjoint and injective. Let $R$ be a bounded self-adjoint operator on $H$ such that $R S=S R$. Prove that $H$ has an orthonormal basis $\left(e_{n}\right)_{n \geqslant 1}$, where, for every $n, e_{n}$ is an eigenvector, both of $S$ and of $R$.

[You may assume, without proof, results about self-adjoint operators on finite-dimensional spaces.]

B4.7

commentThroughout this question, $H$ is an infinite-dimensional, separable Hilbert space. You may use, without proof, any theorems about compact operators that you require.

Define a Fredholm operator $T$, on a Hilbert space $H$, and define the index of $T$.

(i) Prove that if $T$ is Fredholm then $\operatorname{im} T$ is closed.

(ii) Let $F \in \mathcal{B}(H)$ and let $F$ have finite rank. Prove that $F^{*}$ also has finite rank.

(iii) Let $T=I+F$, where $I$ is the identity operator on $H$ and $F$ has finite rank; let $E=\operatorname{im} F+\operatorname{im} F^{*}$. By considering $T \mid E$ and $T \mid E^{\perp}$ (or otherwise) prove that $T$ is Fredholm with ind $T=0$.

(iv) Let $T \in \mathcal{B}(H)$ be Fredholm with ind $T=0$. Prove that $T=A+F$, where $A$ is invertible and $F$ has finite rank.

[You may wish to note that $T$ effects an isomorphism from $(\operatorname{ker} T)^{\perp}$ onto $\operatorname{im} T$; also ker $T$ and $(\operatorname{im} T)^{\perp}$ have the same finite dimension.]

(v) Deduce from (iii) and (iv) that $T \in \mathcal{B}(H)$ is Fredholm with ind $T=0$ if and only if $T=A+K$ with $A$ invertible and $K$ compact.

(vi) Explain briefly, by considering suitable shift operators on $\ell^{2}$ (i.e. not using any theorems about Fredholm operators) that, for each $k \in \mathbb{Z}$, there is a Fredholm operator $S$ on $H$ with ind $S=k$.

B1.10

commentState and prove the Riesz representation theorem for bounded linear functionals on a Hilbert space $H$.

[You may assume, without proof, that $H=E \oplus E^{\perp}$, for every closed subspace $E$ of $H$.]

Prove that, for every $T \in \mathcal{B}(H)$, there is a unique $T^{*} \in \mathcal{B}(H)$ such that $\langle T x, y\rangle=\left\langle x, T^{*} y\right\rangle$ for every $x, y \in H$. Prove that $\left\|T^{*} T\right\|=\|T\|^{2}$ for every $T \in \mathcal{B}(H)$.

Define a normal operator $T \in \mathcal{B}(H)$. Prove that $T$ is normal if and only if $\|T x\|=\left\|T^{*} x\right\|$ for every $x \in H$. Deduce that every point in the spectrum of a normal operator $T$ is an approximate eigenvalue of $T$.

[You may assume, without proof, any general criterion for the invertibility of a bounded linear operator on $H$.]

B3.8

commentLet $T$ be a bounded linear operator on a Hilbert space $H$. Define what it means to say that $T$ is (i) compact, and (ii) Fredholm. What is the index, ind $(T)$, of a Fredholm operator $T$ ?

Let $S, T$ be bounded linear operators on $H$. Prove that $S$ and $T$ are Fredholm if and only if both $S T$ and $T S$ are Fredholm. Prove also that if $S$ is invertible and $T$ is Fredholm then $\operatorname{ind}(S T)=\operatorname{ind}(T S)=\operatorname{ind}(T)$.

Let $K$ be a compact linear operator on $H$. Prove that $I+K$ is Fredholm with index zero.

B4.7

commentWrite an essay on the use of Hermite functions in the elementary theory of the Fourier transform on $L^{2}(\mathbb{R})$.

[You should assume, without proof, any results that you need concerning the approximation of functions by Hermite functions.]