Hilbert Spaces

Prove that the correspondence $L\to \widetilde L$ constructed in 4.1 of Chapter III of the Theory part defines a covariant functor from the category of pre-Hilbert spaces to the category of Hilbert spaces.

Show that $\widetilde L$ is a universal object in a suitable category.

$L^{2}\left({[-1,1],dx}\right)$;

$L^{2}\left({[-1,1],\frac{dx}{\sqrt{1-x^2}}}\right)$;

$L^{2}[{[0,\infty ),e^{-x}\, dx}]$;

$L^{2}\left({(-\infty ,\infty ),e^{-x^2}\, dx}\right)$.

$\displaystyle \left\langle P, Q \right\rangle =\iint_{\left|z\right|\le R}P(z)\overline{Q(z)}\, dx\, dy$;

$\displaystyle \left\langle P, Q \right\rangle =\iint_{\mathbb {C}}P(z)\overline{Q(z)}e^{-\left|z\right|^2}\, dx\, dy$.

is equal to zero for $\left|b-a\right|\le 1$;

is different from zero for $\left|b-a\right|>1$.

Prove that the corresponding Hilbert space $B$ is nonseparable and has the continuum Hilbert basis $\left\{ e^{i\lambda x}\right\}_{\lambda \in \mathbb {R}}$.

Prove that $B$ contains the space of almost periodic continuous functions, i.e., the closure of $B_0$ in the uniform norm $\left\| f\right\| =\sup_{x\in \mathbb {R}}\left|f(x)\right|$, but does not coincide with it.

polynomials;

step functions;

trigonometric polynomials;

functions lying in any given dense subspace.

the polynomials in $x$;

the polynomials in $x^2$;

the polynomials with zero free term;

the polynomials with sum of coefficients equal to zero.

the functions equal to zero for $x\le 0$;

the functions equal to zero at the point $x=0$.

if the chords have a common endpoint and opposite directions with respect to $t$;

if the chords have a common endpoint and the same direction with respect to $t$.

Prove that in every pre-Hilbert space the parallelogram law is satisfied:

Prove that in every normed linear space over $\mathbb {R}$ or $\mathbb {C}$ in which this law is satisfied it is possible to introduce a scalar product such that $\left\| x\right\|^2= \left\langle x, x \right\rangle$.

$\displaystyle \left\langle x, y \right\rangle =\frac1N\sum_{k=1}^N\left\| x+e^{2\pi ik/N}y\right\|^2e^{2\pi ik/N}$ for $N\ge 3$;

$\displaystyle \left\langle x, y \right\rangle =\frac1{2\pi }\int_0^{2\pi }\left\| x+e^{i\theta }y\right\|^2e^{i\theta }\, d\theta$.

Prove that every operator $A$ on a complex Hilbert space $H$ has a unique representation in the form $A=B+iC$, where $B$ and $C$ are Hermitian operators. (This is sometimes written $B=\operatorname {Re}A$, $C=\operatorname {Im}A$.)

Verify that $A$ is normal if and only if $\operatorname {Re}A$ and $\operatorname {Im}A$ commute.

Prove that $A$ is unitary if and only if it is normal and $(\operatorname {Re}A)^2+(\operatorname {Im}A)^2=1$.

every operator $P$ on a Hilbert space having the property that $P^2=P^*=P$ is an orthogonal projection, i.e., a projection onto a closed subspace $H_1$ parallel to its orthogonal complement $H_2$;

every operator $S$ having the property $S^{-1}=S^*=S$ is an orthogonal reflection in some subspace.

Prove that the numerical sequence $a(k)=\ln \left\langle A^kx, x \right\rangle$ is convex, i.e., $a[(k+l)/2]\le [a(k)+a(l)]/2$.

Prove the inequality $\left\| Ax\right\|^2\le Q_A(x)\cdot \left\| A\right\|$, where $Q_A(x)= \left\langle Ax, x \right\rangle$.

$H_1$ is invariant under $A$;

$H_1$ and $H_2$ are invariant under $A$.

Suppose that $\dim L_i=\dim M_i=1$. Prove that the pairs $(L_1,L_2)$ and $(M_1,M_2)$ are congruent in a real space if and only if the angles between the vectors generating these spaces are equal.

Express the angle between the vectors generating the spaces $L_1$ and $L_2$ in terms of the projections $P_1$ and $P_2$.

Find a criterion for the congruence of two pairs of one-dimensional subspaces of a complex space.

Prove that the $\lambda_i$ are real numbers between $0$ and $1$.

Prove that there are not more than $k=\min (\dim L_1,\dim L_2)$ nonzero numbers $\lambda_i$. Arrange them in decreasing order and set $\varphi_i=\arccos \sqrt{\lambda_i}$, $i=1,2,\ldots ,k$. The numbers $\varphi_i$ are called the angles between the subspaces $L_1$ and $L_2$.

If $\dim L_1=1$, $\dim M_1=1$, then there is a unique angle $\varphi_1$ between $L_1$ and $L_2$ and a unique angle $\psi_1$ between $M_1$ and $M_2$. Prove that the pairs $(L_1,L_2)$ and $(M_1,M_2)$ are congruent if and only if $\varphi_1=\psi_1$.

Prove the following general criterion for congruence: The pair $(L_1,L_2)$ is congruent to the pair $(M_1,M_2)$ if and only if the angles between $L_1$ and $L_2$ are equal to the corresponding angles between $M_1$ and $M_2$.

The opening between the subspaces $L_1$ and $L_2$ is defined to be the number $\left\| P_1-P_2\right\|$. Express this number in terms of the angles between $L_1$ and $L_2$.

$(\operatorname {im}A)^\perp =\ker A^*$;

$(\ker A)^\perp =\overline{\operatorname {im}A^*}$

Prove that every positive operator $A$ on a Hilbert space has a positive square root $B$, which can be obtained as the strong limit of the sequence $B_n$ determined by the initial condition $B_0=0$ and the recursion formula $B_{n+1}=B_n+[(A-B_n^2)/(2\sqrt{\left\| A\right\| })]$.

Prove that the positive square root $B$ of a positive operator $A$ is unique.

Prove that every operator $A$ on a finite-dimensional Hilbert space $H$ admits representations in the form $A=RU$ and $A=VS$, where $R$ and $S$ are positive operators, and $U$ and $V$ are unitary operators. These expressions are called polar decompositions of $A$. (In the case $\dim H=1$ this reduces to the expression of a complex number $a$ in the form $re^{i\varphi }$.)

Prove that $R$ and $S$ are uniquely determined by $A$. Is this true for $U$ and $V$?

Verify that the one-sided shift operator $T$ on $l_2$ does not admit a polar decomposition in the sense of part (a).

Prove that if $A$ is a Hilbert--Schmidt operator, then the quantity $\left\| A\right\|_{2}=(\sum_{\beta \in B}\left\| Ax_\beta \right\| )^{1/2}$ does not depend on the choice of basis and defines in the space $\mathcal L_2(H)$ of Hilbert--Schmidt operators a norm which majorizes the usual operator norm.

Prove that the norm $\left\| \cdot \right\|_{2}$ is generated by the scalar product $\langle A,B\rangle_{\mathcal L_2(H)}=\sum_{\gamma \in \Gamma }(Ay_\gamma ,By_\gamma )_H$, where $\left\{ y_\gamma \right\}_{\gamma \in \Gamma }$ is any Hilbert basis in $H$.

Prove that every Hilbert--Schmidt operator is compact.

Construct an isomorphism between $\mathcal L_2(H)$ and the Hilbert tensor product of $H$ and $H'$.

Prove that every Hilbert--Schmidt operator on $L^{2}\left({X,\mu }\right)$ is an integral operator with kernel $K\inL^{2}\left({X\times X,\mu \times \mu }\right)$.

if $A$ is a nuclear operator, then for any basis $\left\{ y_\gamma \right\}_{\gamma \in \Gamma }$ in $H$ the series $\sum_\gamma \left\langle Ay_\gamma , y_\gamma \right\rangle$ converges, and its sum does not depend on the choice of basis (it is denoted by $\operatorname {tr}A$ and called the trace of the operator $A$);

if $A$ is a nuclear operator and $B$ is a bounded operator, then $AB$ and $BA$ are nuclear, and $\operatorname {tr}AB=\operatorname {tr}BA$;

the collection of all nuclear operators forms a Banach space $\mathcal L_1(H)$ with respect to the norm $\left\| \cdot \right\|_{1}$ defined by the formula $\left\| A\right\| =\operatorname {tr}R$, where $A=RU$ is the polar decomposition of $A$;

the isomorphisms $\mathcal K(H)'\simeq \mathcal L_1(H)$, $\mathcal L_1(H)'\simeq \mathcal L(H)$ are valid, where $\mathcal K(H)$ is the space of compact operators on $H$ with the usual norm.

Construct continuous bases in the spaces of Problem 465.

Prove that the mapping $\xi \mapsto \left\langle \xi , \xi_x \right\rangle$ is an isometry of $H$ into $L^{2}\left({X,\mu }\right)$.

Prove that $\dim H=\mu (X)$.

Prove that $\operatorname {tr}A=\int_X \left\langle A\xi_x, \xi_x \right\rangle \, d\mu (x)$ for a nuclear operator $A$.