Spectral theorem in terms of the Lebesgue–Stieltjes integral

Let $A$ be a self-adjoint operator in a Hilbert space $H$ and let $f, g \in C(\mathbb {R})$ be two functions that coincide identically on the ray $t \leqslant \lambda$. Prove that $(f(A)-g(A)) E_{\lambda } = 0$.

Let $A$ be a self-adjoint operator in a Hilbert space $H$, and $E_{\lambda }$ the resolution of the identity for the operator $A$. Prove the following properties:

Let $A$ be a self-adjoint operator in a Hilbert space $H$. Prove that the family of subspaces $H_{\lambda }$ associated with the operator $A$ has the following properties:

Let $A$ be a self-adjoint operator in a Hilbert space $H$. Prove that:

Let $A$ be a self-adjoint operator in a Hilbert space $H$. Prove that if $\lambda \in \sigma_{\text{ess }}(A)$ (see Definition 12.4) or $\lambda \in \sigma_{\mathrm{c}}(A)$, then there exists an orthonormal Weyl sequence for the operator $A-\lambda I$ (compare with problem 12.30).

Definition 12.4: A number $\lambda$ belongs to the essential spectrum of the operator $A$ ($\lambda \in \sigma_{\text{ess }}(A)$) if the operator $A-\lambda I$ is not Fredholm. The remaining points of the spectrum belong to the inessential spectrum.

Let $A$ be a self-adjoint operator in a Hilbert space $H$. Prove that

Prove that if $\lambda \in \sigma (A) \backslash \sigma_{\mathrm{ess}}(A)$, then $\lambda$ is an isolated eigenvalue of the operator $A$ of finite multiplicity.

Find the family of projectors $E_{\lambda }$ and subspaces $H_{\lambda }$ associated with the operator $P$ of orthogonal projection onto some subspace $Y \subset H$.

Find the family of projectors $E_{\lambda }$ and subspaces $H_{\lambda }$ associated with the operator $A$, given in $l_{2}(n)$ by the matrix

where all the numbers $\lambda_{i}, i = 1, \ldots , n$, are real and pairwise distinct. How does the answer change if some of the numbers $\lambda_{i}$ coincide?

Find the family of projectors $E_{\lambda }$ and subspaces $H_{\lambda }$ associated with the operator $A: L_{2}[0,1] \rightarrow L_{2}[0,1],(A x)(t) = t x(t)$.

Find the family of projectors $E_{\lambda }$ and subspaces $H_{\lambda }$ associated with the operator $A\left(x_{1}, x_{2}, \ldots \right) = \left(\lambda_{1} x_{1}, \lambda_{2} x_{2}, \ldots \right)$ in the space $l_{2}$, where $\left\{ \lambda_{k}\right\}_{1}^{\infty }$ is a bounded real sequence.

Find the family of projectors $E_{\lambda }$ and subspaces $H_{\lambda }$ associated with the operator given by the matrix:

Let $a$ be a continuous strictly increasing function on $[0,1]$. Find the family of projectors $E_{\lambda }$ and subspaces $H_{\lambda }$ associated with the operator $A: L_{2}[0,1] \rightarrow L_{2}[0,1], A x(t) = a(t) x(t)$.

Describe the family of projectors $E_{\lambda }$ and subspaces $H_{\lambda }$ associated with a self-adjoint compact operator $A$ in a Hilbert space (in terms of eigenvalues and eigenvectors).

Find the family of projectors $E_{\lambda }$ and subspaces $H_{\lambda }$ associated with the operator $A \in \mathcal{B}\left(L_{2}[-1,1]\right),(A x)(t) = t^{2} x(-t)$.

Let $A$ be a bounded operator in a Hilbert space $H$, and suppose that for every orthonormal basis $\left\{ e_{n}\right\}_{1}^{\infty }$ in $H$ we have $\left\| A e_{n}\right\| \rightarrow 0$ as $n \rightarrow \infty$. Prove that the operator $A$ is compact (cf. problems 11.54 and 11.45).

Prove that any self-adjoint operator $A$ in a Hilbert space $H$ can be approximated to any accuracy by finite linear combinations of orthogonal projectors in the space $\mathcal{B}(H)$.

Let $A$ be a self-adjoint operator in a Hilbert space $H$, and $B$ a self-adjoint operator in a Hilbert space $K$. Let $H_{\lambda }$ and $K_{\lambda }$ be the families of subspaces associated with the operators $A$ and $B$. Prove that the operators $A$ and $B$ are similar (unitarily equivalent), i.e., $B C = C A$ for some $C \in \mathcal{I B}(H, K)$ ($C$ a unitary operator from $H$ to $K$), if and only if $C$ realizes a bijection (unitary isomorphism) of $H_{\lambda }$ and $K_{\lambda }$ for every $\lambda \in \mathbb {R}$.

Let $U$ be a unitary operator in a Hilbert space $H$. Prove that there exists a unique projector-valued measure $E_{\lambda }$ on the circle $\mathbb {T} = \left\{ z \in \mathbb {C}:\left|z\right| = 1\right\}$ with values in $\mathcal{B}(H)$, for which the equality

holds for every bounded Borel function $f$ on $\mathbb {T}$.

Let $U$ be a unitary operator in a Hilbert space $H$. Prove that $\frac{1}{n} \sum_{k = 1}^{n} U^{k} \xrightarrow {s} P$, where $P$ is the orthoprojector onto the subspace $\operatorname {Ker}(U-I)$.

Let $A$ and $B$ be self-adjoint operators in a Hilbert space $H$ and $E_{\lambda }, F_{\mu }$ their resolutions of the identity respectively. Prove that $A$ and $B$ commute if and only if $E_{\lambda } F_{\mu } = F_{\mu } E_{\lambda }$ for any $\lambda , \mu \in \mathbb {R}$.

Let $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ be complete separable metric spaces, $\Sigma_{1}$ and $\Sigma_{2}$ the $\sigma$-algebras of Borel sets on these spaces. Suppose that on $(\mathcal{M}_{1}, \Sigma_{1})$ and $(\mathcal{M}_{2}, \Sigma_{2})$ there are given, respectively, Borel resolutions of the identity $E^{(1)}$ and $E^{(2)}$, whose values are orthogonal projectors in one and the same Hilbert space $H$. Suppose these resolutions of the identity commute, i.e., for all $\Omega_{i} \in \Sigma_{i}(i = 1,2) E^{(1)}\left(\Omega_{1}\right) E^{(2)}\left(\Omega_{2}\right) = E^{(2)}\left(\Omega_{2}\right) E^{(1)}\left(\Omega_{1}\right)$. On the Cartesian product $\mathcal{M}: = \mathcal{M}_{1} \times \mathcal{M}_{2}$ define $\Sigma$ — the Borel $\sigma$-algebra of subsets containing all "rectangles" of the form $\Omega_{1} \times \Omega_{2}$. Prove that on $(\mathcal{M}, \Sigma )$ there exists a unique resolution of the identity, determined by the condition

where $\Omega_{i} \in \Sigma_{i}, i = 1,2$.

Let $A = A_{\mathrm{re}}+i A_{\mathrm{im}}$ be a bounded normal operator in a Hilbert space $H, \Pi : = \sigma \left(A_{\mathrm{re}}\right)+i \sigma \left(A_{\mathrm{im}}\right)$. Prove that there exists a resolution of the identity $E$, defined on the Borel subsets of П, such that $A = \int_{\Pi } \lambda d E_{\lambda }$, where the integral is understood in the Lebesgue-Stieltjes sense (see the preamble of this section).

Let $A$ be a bounded normal operator in a Hilbert space $H$. Prove that

Let $A = A_{\mathrm{re}}+i A_{\mathrm{im}}$ be a bounded normal operator in a Hilbert space $H$. Prove that $\sigma (A) \subseteq \sigma \left(A_{\mathrm{re}}\right) + i \sigma \left(A_{\mathrm{im}}\right)$. Give examples of normal operators for which

Let $A$ be a unitary operator in a Hilbert space $H$. Prove that $\sigma (A) \subseteq \sigma \left(A_{\mathrm{re}}\right)+i \sigma \left(A_{\mathrm{im}}\right)$.