Spectral Decomposition of Operators

Let $A$ be a selfadjoint operator in a finite-dimensional space. Reduce it to the form of multiplication by a function.

The operator $A$ consists in multiplication by the function $a(x)=x^2$ in $H=L^{2}\left({[-1,1]}\right)$. Prove that:

Prove that if $A$ is an operator with simple spectrum in an infinite-dimensional Hilbert space $H$, then the operators $1,A,\ldots ,A^n$ are linearly independent for any $n$.

Suppose that $B$ and $C$ are bounded selfadjoint operators that commute. Prove that there exists a bounded selfadjoint operator $A$ such that $B$ and $C$ are functions of $A$.

Let $A$ be a selfadjoint operator with simple spectrum. Prove that every bounded operator $B$ that commutes with $A$ is a function of $A$.

Prove that $\left\| f(A)\right\| =\sup_{t\in \sigma (A)}\left|f(t)\right|$ for any bounded Borel function of a selfadjoint operator $A$.

Reduce the convolution operator $S(f)$, $f\in L^{1}\left({\mathbb {R},dx}\right)$, to the form of multiplication by a function. For what functions $f$ is this operator selfadjoint?

Can the convolution operator $S(f)$, $f\in L^{1}\left({\mathbb {R},dx}\right)$, be unitary?

Let $G$ be a locally compact abelian group with invariant measure $\mu$. Determine conditions on a function $f\in L^{1}\left({G,\mu }\right)$ under which the convolution operator $S(f)$ is:

Let $A$ be the integral operator on $L^{2}\left({0,1}\right)$ given by the formula $Af(x)=\int_0^1\min (x,y)f(y)\, dy$. Reduce this operator to the form of multiplication by a function.

Let $A$ be an unbounded selfadjoint operator in $H$, and $\Gamma_A$ the graph of $A$ in $H\oplus H$. Prove that the orthogonal complement of $\Gamma_A$ in $H\oplus H$ is $\tau (\Gamma_A)$.

Under the conditions of Problem 696 denote the projection of a vector $x\oplus 0$ on $\Gamma_A$ by $(y\oplus Ay)$ and its projection on $\Gamma_A^\perp$ by $(-Az)\oplus z$. Prove that:

Prove that the finite-difference operator $\Delta_h\varphi (x)=(1/h)[\varphi (x+h)-\varphi (x)]$ is a function of the differentiation operator.

Let $f$ be a continuous function on the circle. Give an explicit expression for the operator $f(F)$, where $F$ is the Fourier operator $F\varphi (y)=\int_{\mathbb {R}} e^{-2\pi ixy}\varphi (y)\, dy$.

Let $\lambda$ be a projection-valued measure on $[a,b]$ with values in $\operatorname {End} H$, and $f$ a continuous function on $[a,b]$. For each partition $T=\left\{ a=t_0\leq t_1\leq \cdots \leq t_n=b\right\}$ and each tuple of points $\xi =\left\{ \xi_k\right\}$, $\xi_k\in [t_k,t_{k+1}]$, define the Riemann integral sum

Prove the following properties of the integral of bounded functions with respect to a projection-valued measure:

The definition referred to here (Ch. V, 2.2) is: a mapping $\lambda :B\to \operatorname {End}H$, where $B$ is a $\sigma$-algebra on $X$, is a projection-valued measure if:

1. $\lambda \left(E\right)=\lambda \left(E\right)^*$ for every $E\in B$;

2. $\lambda \left(E_1\cap E_2\right)=\lambda \left(E_1\right)\lambda \left(E_2\right)$ for $E_1,E_2\in B$;

3. $\lambda \left(E_1\cup E_2\right)=\lambda \left(E_1\right)+\lambda_2\left(E_2\right)$ for disjoint $E_1,E_2\in B$;

4. if $\lim_{n\to \infty }E_n=E$ exists, then $\operatorname {s-lim}_{n\to \infty }\lambda \left(E_n\right)=\lambda \left(E\right)$.

Suppose that $H_1=L^{2}\left({[0,1]}\right)$, $H_2=L^{2}\left({[0,1]\times [0,1]}\right)$. Define projection-valued measures $\lambda_1$ and $\lambda_2$ with values in $\operatorname {End} H_1$ and $\operatorname {End} H_2$, respectively, by setting $\lambda_i\left(E\right)=M(\chi_E(x))$. Does there exist an isomorphism $U:H_1\to H_2$ under which $\lambda_1$ passes into $\lambda_2$?

Suppose that $X$ is a set with a $\sigma$-algebra $B$, $H$ is a Hilbert space, and for each $\xi \in H$ a finite measure $\mu_\xi$ is given on $(X,B)$ such that:

1. $\mu_\xi \left(X\right)=\left\| \xi \right\|^2$;

2. $\mu_{\xi -\eta }+\mu_{\xi +\eta }=2\mu_\xi +2\mu_\eta$. Does there exist a projection-valued measure on $X$ with values in $\operatorname {End} H$ such that $\mu_\xi =\lambda_\xi$ for all $\xi \in H$?

Here $\lambda_\xi \left(E\right)= \left\langle \lambda \left(E\right)\xi , \xi \right\rangle$.

Let $X$ be a compact metric space, and $H$ a Hilbert space. A representation of the algebra $C(X)$ in $H$ is defined to be a mapping $\varphi :C(X)\to \operatorname {End} H$ having the following properties:

1. $\varphi$ is an algebra homomorphism;

2. $\varphi (\overline{f})=\varphi (f)^*$;

3. $\varphi (1)=1$ (the $1$ on the left is the function on $X$ and that on the right is the identity operator on $H$). Prove that there exists a unique projection-valued measure $\lambda$ on $X$ with values in $\operatorname {End} H$ such that $\varphi (f)=\int_Xf(x)\, d\lambda (x)$.

Let $A$ be a selfadjoint operator in a Hilbert space $H$, $\lambda$ its spectral measure, and $E$ a Borel set on the line. Denote by $H_E$ the subspace $\lambda \left(E\right)H$. Prove that:

Prove that the spectrum of a selfadjoint operator $A$ consists of precisely those points $a\in \mathbb {R}$ such that $\lambda \left((a-\varepsilon ,a+\varepsilon )\right)\ne 0$ for any $\varepsilon >0$. (Here $\lambda$ is the spectral measure of $A$.)

Prove that a point $a$ belongs to the spectrum of a selfadjoint bounded operator $A$ on a Hilbert space $H$ if and only if there exists a sequence of unit vectors $\xi_n\in H$ such that $\left\| A\xi_n-a\xi_n\right\| \to 0$ as $n\to \infty$.

The definition of the essential spectrum of a bounded selfadjoint operator $A$ is obtained from the Weyl criterion (see Problem 708) by imposing the additional condition that the sequence $\left\{ \xi_n\right\}$ be orthonormal. Prove that if $B$ is a selfadjoint compact operator, then the essential spectra of $A$ and $A+B$ coincide.

Suppose that $A$ is a bounded selfadjoint operator with spectral measure $\lambda$. Prove the equality

Suppose that $U$ is a unitary operator in a Hilbert space $H$. Prove that there exists a unique Borel projection-valued measure $\lambda$ on the circle $\mathrm T$ with values in $\operatorname {End} H$ such that

for any bounded Borel function $f$ on $\mathrm T$.

Let $U$ be a unitary operator on a Hilbert space $H$. Prove that $\operatorname {s-lim}_{N\to \infty }(1/N)\sum_{k=1}^NU^k$ exists and is equal to the orthogonal projection onto $\ker (U-1)$.

Let $A$ be any bounded operator, and $f$ an analytic function in a domain $\Omega$ containing $\sigma (A)$. Define the operator $f(A)$ by

where $C$ is any contour in $\Omega$ encompassing $\sigma (A)$.

Prove that any family of pairwise commuting bounded selfadjoint operators on a separable Hilbert space can be simultaneously reduced to the form of multiplication by a function.

Let $A$ be a selfadjoint operator in a Hilbert space $H$, $\lambda$ its spectral measure, and $f$ a Borel function on $\mathbb {R}$. Define $f(A)$ by the formula

on the subspace $D_A\subset H$ consisting of the vectors $\xi \in H$ for which

Prove that:

Find the operator $A$ in the representation $V(t)=e^{itA}$ for the one-parameter group in $L^{2}\left({\mathbb {R},dx}\right)$ defined by $V(t)f(\tau )=f(t+\tau )$.

Suppose that the one-parameter group $U(t)$ of unitary operators in $H$ has the property $U(1)=1$. Prove that $U(t)=e^{2\pi itA}$, where $\sigma (A)\subset \mathbb {Z}$.

Does there exist a one-parameter group $V(t)$ such that $V(1)=U$ for any given unitary operator $U$?

Find the spectral decomposition of the selfadjoint extension of the operator $A=-(d^2/dx^2)+x^2$ with initial domain $D(A)=S(\mathbb {R})$.