Functional calculus and the spectral theorem

(Spectral mapping theorem for polynomials). Prove that for a bounded operator $A$ in a Banach space and for a polynomial $p$ the relation $\sigma (p(A)) = p(\sigma (A))$ holds.

Let $X$ be a Banach space, $A \in \mathcal{B}(X)$ and $\left\| A\right\| \in \sigma (A)$. Prove that $\left\| I+A\right\| = 1+\left\| A\right\|$.

Let $A$ be a bounded operator in a Banach space $X$, and let $r(z) = \frac{p(z)}{q(z)}, z \in \mathbb {C}$, where $p$ and $q$ are polynomials, and suppose $q(z)$ has no zeros on the spectrum of the operator $A$. Prove that for any $z \in \sigma (A)$ the relation

holds (here $r(A): = p(A)(q(A))^{-1}$). The next step is to define a holomorphic function $f$ of the operator $A$. Note that in this case the domain of holomorphy of the function $f$ must contain $\sigma (A)$.

Prove Theorem 13.1.

Theorem 13.1 (spectral theorem in terms of the holomorphic calculus, N. Dunford, 1943): Let $X$ be a Banach space, $A \in \mathcal{B}(X)$, $U_{r} = \left\{ z \in \mathbb {C}: \left|z\right| < r\right\}$ (in particular, if $r = \infty$, then $U_{r} = \mathbb {C}$). Let $\left\| A\right\| < r$. Then there exists a unique continuous homomorphism $\varphi : \mathcal{A}\left(U_{r}\right) \rightarrow \mathcal{B}(X)$ assigning to every holomorphic function $f \in \mathcal{A}\left(U_{r}\right)$ an operator $f(A) \in \mathcal{B}(X)$ and satisfying properties (1)--(4):

1. $\varphi (1) = I$ (here $1$ is the function identically equal to one on the disk $U_{r}$);

2. $\varphi (z) = A$;

3. $\varphi (f+g) = \varphi (f)+\varphi (g)$, $\varphi (f g) = \varphi (f) \circ \varphi (g)$, $f, g \in \mathcal{A}\left(U_{r}\right)$ (the homomorphism conditions);

4. $\varphi$ is a continuous linear operator from $\mathcal{A}\left(U_{r}\right)$ to $\mathcal{B}(X)$ (that is, if a sequence of functions $\left\{ f_{n}\right\}_{1}^{\infty }$ from $\mathcal{A}\left(U_{r}\right)$ converges to $f \in \mathcal{A}\left(U_{r}\right)$ uniformly on every compact subset of $U_{r}$, then $f_{n}(A) \rightrightarrows f(A)$ in $\mathcal{B}(X)$). This homomorphism satisfies the additional condition: (5) if some bounded operator $B$ commutes with $A$, then it also commutes with every operator $f(A)$.

(Spectral mapping theorem). Let $X$ be a Banach space, $A \in \mathcal{B}(X)$. Let $f(z)$ be a function holomorphic on $\mathbb {C}$ (an entire function). Prove that $\sigma (f(A)) = f(\sigma (A))$.

Let $X$ be a Banach space, $A \in \mathcal{B}(X)$ and $\lambda_{0} \in \sigma \left(A^{2}\right)$. Is it true that $\sqrt{\lambda_{0}} \in \sigma (A)$ (for at least one value of the root)? Is this statement true with $\sigma$ replaced by $\sigma_{\mathrm{p}}, \sigma_{\mathrm{c}}$ or $\sigma_{\mathrm{r}}$?

Find the spectrum of the operator $A$, acting by the rule $A x = \left(x_{1}+x_{2},-x_{2}+x_{3}, x_{3}+x_{4},-x_{4}+x_{5}, \ldots \right)$ in the space $l_{p}, p \in [1, \infty ]$ (classify the spectrum, determining the point, continuous, and residual spectrum).

Find the operator $f(A)$, where $f$ is an entire function, $A \in \mathcal{B}\left(L_{p}[0,1]\right)$, $p \in [1, \infty ],(A x)(t) = a(t) x(t)$, where $a \in L_{\infty }[0,1]$.

Let $X$ be a Banach space, $A \in \mathcal{B}(X)$. Prove that the operator-valued function $U(t) = \exp (t A), t \in \mathbb {C}$, is differentiable and $U^{\prime }(t) = A U(t) = U(t) A$.

Let $X$ be a Banach space, $A \in \mathcal{B}(X)$. Prove that $\left\| \exp (A)\right\| \leqslant e^{\left\| A\right\| }$. Prove that $\exp (A)$ is an invertible operator and its inverse is $\exp (-A)$.

Let $A$ be the integration operator:

considered in

Let $X$ be a Banach space, $A \in \mathcal{B}(X)$. Do the operators $\left(I+\frac{1}{n} A\right)^{n}$ converge (uniformly) to the operator $\exp (A)$ as $n \rightarrow \infty$?

Let $X$ be a Banach space, $A, B \in \mathcal{B}(X)$. Prove that if $A B = B A$, then $\exp (A+B) = \exp (A) \exp (B)$. Give an example of non-commuting operators $A$ and $B$ for which $\exp (A+B) \neq \exp (A) \exp (B)$.

Let $X$ be a Banach space, $A \in \mathcal{B}(X)$. Is it true that $(\sin (A))^{2}+(\cos (A))^{2} = I$?

Let $X$ be a Banach space, $A \in \mathcal{B}(X)$. Prove that if $\lambda \in \sigma_{\mathrm{p}}(A)$, then $e^{\lambda } \in \sigma_{\mathrm{p}}(\exp (A))$. Let $\mu \in \sigma_{\mathrm{p}}(\exp (A))$ and let $\lambda \in \mathbb {C}$ be such that $e^{\lambda } = \mu$. Is it true that $\lambda \in \sigma_{\mathrm{p}}(A)$?

Can the shift operator $A \in \mathcal{B}\left(L_{2}(\mathbb {R})\right)$, $A x(t) = x(t+a), a \neq 0$, be represented in the form $A = \exp (B)$?

Prove that for a self-adjoint operator $A$ in a Hilbert space and a polynomial $p(x)$

Give an example showing that this statement is false for an arbitrary non-self-adjoint operator. Prove this statement for an arbitrary normal operator $A$ (use problem 12.35).

Let $A = A^{*}$ and $\sigma (A) = \left\{ 0 ; 1\right\}$. Prove that the operator $A$ is an orthoprojector.

Let $A = A^{*}$ and $\sigma (A) = \left\{ -1 ; 1\right\}$. Find $A^{-1}$.

Let $H$ be a Hilbert space, $A$ a self-adjoint operator and $\lambda \in \rho (A)$. Prove that the estimate $\left\| R_{\lambda }(A)\right\| \leqslant \frac{1}{\operatorname {dist}(\lambda , \sigma (A))}$ holds:

Let $H$ be a Hilbert space, $A$ a self-adjoint operator. Prove that its Cayley transform

is defined and is a unitary operator. Prove that $A = i(I+U)(I-U)^{-1}$.

Let $H$ be a Hilbert space, $A$ a self-adjoint operator in $H$. Set $U(t): = \exp (i t A) = \sum_{k = 0}^{\infty } \frac{(i t A)^{k}}{k!}, t \in \mathbb {C}$. Prove that $U(t)$ is a holomorphic operator group with generator $A$.

Let $H$ be a Hilbert space, $A$ a self-adjoint operator in $H, \varphi_{[a, b]}$ and $\varphi_{[c, d]}$ the functional calculi for continuous functions on the segments $[a, b]$ and $[c, d]$, where $a \leqslant c < d \leqslant b$ and $\sigma (A) \subset [c, d], \tau : C[a, b] \rightarrow C[c, d]$ the restriction operator. Prove that $\varphi_{[a, b]} = \varphi_{[c, d]} \tau$, i.e., the functional calculi on the indicated segments are consistent.

Let $H$ be a Hilbert space, $A$ a self-adjoint operator in $H$. Prove the consistency of the functional calculi for continuous and holomorphic functions. Namely, if $f$ is a function holomorphic in a disk containing $\sigma (A)$, then the operator $f(A)$, constructed according to Theorem 13.1, coincides with the operator constructed according to Theorem 13.2.

Theorem 13.1 (N. Dunford, 1943): Let $X$ be a Banach space, $A \in \mathcal{B}(X)$, $U_{r} = \left\{ z \in \mathbb {C}: \left|z\right| < r\right\}$, $\left\| A\right\| < r$. Then there exists a unique continuous homomorphism $\varphi : \mathcal{A}\left(U_{r}\right) \rightarrow \mathcal{B}(X)$ assigning to every holomorphic function $f \in \mathcal{A}\left(U_{r}\right)$ an operator $f(A) \in \mathcal{B}(X)$ and satisfying: (1) $\varphi (1) = I$; (2) $\varphi (z) = A$; (3) $\varphi (f+g) = \varphi (f)+\varphi (g)$, $\varphi (f g) = \varphi (f) \circ \varphi (g)$; (4) if $f_{n} \rightarrow f$ in $\mathcal{A}\left(U_{r}\right)$ uniformly on every compact subset of $U_{r}$, then $f_{n}(A) \rightrightarrows f(A)$ in $\mathcal{B}(X)$.

Theorem 13.2 (D. Hilbert, 1906; F. Riesz, 1910): For every self-adjoint operator $A$ in a Hilbert space $H$ there exists a unique continuous homomorphism $\varphi$ of the algebra $C(\sigma (A))$ into $\mathcal{B}(H)$ with the properties: (1) $\varphi (f(x) \equiv 1) = I$; (2) $\varphi (f(x) \equiv x) = A$; (3) $\varphi (f+g) = \varphi (f)+\varphi (g)$, $\varphi (f g) = \varphi (f) \circ \varphi (g)$; (4) if $f_{n} \rightarrow f$ in $C(\sigma (A))$, then $f_{n}(A) \rightrightarrows f(A)$.

Let $H$ be an infinite-dimensional Hilbert space, $A$ a self-adjoint operator. Prove that if $A$ is compact, and $f$ is a function defined and continuous on $\sigma (A)$, then the operator $f(A)$ is compact if and only if $f(0) = 0$ (compare with problem 11.19).

Let $f \in C(\mathbb {R})$ and $H$ a Hilbert space. Find $f(A)$ in the case:

Let $A$ be a self-adjoint compact operator in a Hilbert space $H$ and $f \in C(\mathbb {R})$. Find the matrix of the operator $f(A)$ in the basis of eigenvectors of the operator $A$ (see Theorem 12.3).

Theorem 12.3 (D. Hilbert, 1906; E. Schmidt, 1907): Let $H$ be an infinite-dimensional separable Hilbert space and $A = A^{*} \in \mathcal{K}(H)$. Then there exists an orthonormal basis $\left\{ e_{n}\right\}_{n = 1}^{\infty }$ of $H$ consisting of eigenvectors of $A$: $A e_{n} = \lambda_{n} e_{n}$. The nonzero eigenvalues are real and can be numbered in nonincreasing order of absolute value: $\left|\lambda_{1}\right| \geqslant \left|\lambda_{2}\right| \geqslant \ldots \geqslant \left|\lambda_{n}\right| \geqslant \ldots$, and moreover $\left|\lambda_{1}\right| = \left\| A\right\|$ and $\lim_{n \rightarrow \infty } \lambda_{n} = 0$.

Let $H$ be a Hilbert space, $A$ a bounded self-adjoint operator in $H, \sigma (A) \subset [a, b]$ and $f \in C[a, b]$. Prove that for the functional calculus constructed from the operator $A$ on the segment $[a, b]$ the spectral mapping theorem holds: $\sigma (f(A)) = f(\sigma (A))$.

Let $A$ and $B$ be similar self-adjoint operators in $\mathcal{B}(H)$, where $H$ is a Hilbert space (i.e., $A C = C B$ for some $C \in \mathcal{I B}(H)$). Let the segment $[a, b]$ contain the spectra of the operators $A$ and $B$. Prove that for every $f \in C[a, b]$ the operators $f(A)$ and $f(B)$ are similar, and the similarity is realized by means of the same operator $C$.

Let $A \geqslant 0$ in a Hilbert space $H$, and let the function $f$, continuous on the segment $[a, b] \supseteq \sigma (A)$, also be nonnegative. Prove that $f(A) \geqslant 0$.

Let bounded self-adjoint operators $A$ and $B$, acting in Hilbert spaces $H_{1}$ and $H_{2}$ respectively, be similar. Prove that they are unitarily equivalent, i.e., the operator $U \in \mathcal{B}\left(H_{1}, H_{2}\right)$ realizing the similarity $(U A = B U)$ is an isometric isomorphism.

Let $A$ be a self-adjoint operator in a Hilbert space $H$, and let the segment $[a, b]$ contain $\sigma (A)$. Prove that

Let $A$ be a self-adjoint operator in a Hilbert space $H$. Prove that $\left|A\right| = \sqrt{A^{2}}$. Prove that the polar decomposition of a self-adjoint operator in $H$ has the form $A = \left|A\right| W = W\left|A\right|$, where $W$ is a partial isometry (see Definitions 8.8 and 8.6).

Definition 8.6: An operator $W \in \mathcal{B}(H, K)$, where $H$ and $K$ are Hilbert spaces, is called a partially isometric operator (partial isometry) if $H = H_{1} \oplus_{\perp } H_{2}$, $K = K_{1} \oplus_{\perp } K_{2}$, $H_{2} = \operatorname {Ker} W$, and $W: H_{1} \rightarrow K_{1}$ is an isometric isomorphism.

Definition 8.8: The representations $A = W S = R W$ (see problem 8.68: $W$ is a partially isometric operator, $S$ and $R$ are nonnegative operators) are called the polar decomposition of the operator $A$.

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 $H, f \in C[a, b]$ a real-valued function. Prove that ¹

Let $A$ be a self-adjoint operator in a Hilbert space $H, f \in C[a, b]$ a real-valued function. Prove that

Construct the functional calculus (for continuous functions) of a unitary operator $U$ in a Hilbert space $H$. Namely, prove that there exists a unique continuous homomorphism $\varphi$ of the algebra $C(\sigma (U))$ into $\mathcal{B}(H)$, having properties (1)-(4):

1. $\varphi (f(x) \equiv 1) = I$;

2. $\varphi (f(x) \equiv x) = U$;

3. $\varphi (f+g) = \varphi (f)+\varphi (g), \varphi (f g) = \varphi (f) \circ \varphi (g)$ (homomorphism conditions);

4. $\varphi$ is a linear continuous operator from $C(\sigma (U))$ into $\mathcal{B}(H)$ (i.e., if a sequence of functions $\left\{ f_{n}\right\}_{1}^{\infty }$ from $C(\sigma (U))$ converges to $f \in C(\sigma (U))$, then $f_{n}(U) \rightrightarrows f(U)$). This homomorphism also has the properties

5. $(\varphi (f))^{*} = \varphi (g)$, where $g(x) = \bar{f}(1 / x)$;

6. $\left\| \varphi (f)\right\|_{\mathcal{B}(H)} = \max_{t \in \sigma (U)}\left|f(t)\right|$ (a strengthening of the continuity property of the homomorphism $\varphi$);

7. if some bounded operator $V$ commutes with $U$, then it also commutes with every operator $f(U)$.

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