Functional calculus technique

For $T$ a bounded operator on a Banach space, define $e^{T}$ by

Suppose $T$ is a self-adjoint compact operator on a Hilbert space and $\left\| T\right\| \leq \frac{1}{4}$. Prove that there exists a self-adjoint compact operator $S$ such that $S^{2}+S = T$.

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 $\sigma (f(A)) = f(\sigma (A))$.

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

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 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 $H, f \in C[a, b]$ 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 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).

Let $\mathcal{A} = C(X)$, $X$ compact (see Example 3.2). If $g \in C(X)$ and $f \in \text{Hol}(g)$, show that $f(g) = f \circ g$.

Let $a$ be a nilpotent element of $\mathcal{A}$. For $f, g$ in $\text{Hol}(a)$, give a necessary and sufficient condition on $f$ and $g$ that $f(a) = g(a)$.

Let $d \ge 1$ and let $A \in M_d(\mathbb {C})$. Give a necessary and sufficient condition on $f$ in $\text{Hol}(A)$ such that $f(A) = 0$.

If $\mathcal{A}$ is a Banach algebra with identity, $a \in \mathcal{A}$, $f \in \text{Hol}(a)$, and $g$ is analytic in a neighborhood of $f(\sigma (a))$, then $g \circ f \in \text{Hol}(a)$ and $g(f(a)) = g \circ f(a)$.

If $\mathcal{X}$ is a Banach space, $A \in \mathcal{B}(\mathcal{X})$, and $\mathcal{M} \le \mathcal{X}$ such that $(A - \alpha )^{-1} \mathcal{M} \subseteq \mathcal{M}$ for all $\alpha$ in $\rho (A)$, show that $f(A) \mathcal{M} \subseteq \mathcal{M}$ whenever $f \in \text{Hol}(A)$.

If $\mathcal{X}$ is a Banach space, $A \in \mathcal{B}(\mathcal{X})$, and $f \in \text{Hol}(A)$, show that $f(A)^* = f(A^*)$. (See (6.1) below.)

If $\mathcal{H}$ is a Hilbert space, $A \in \mathcal{B}(\mathcal{H})$, and $f \in \text{Hol}(A)$, show that $f(A)^* = \tilde{f}(A^*)$, where $\tilde{f}(z) = \overline{f(\bar{z})}$. (See (6.1) below.)

If $\mathcal{A}$ is a $C^*$-algebra with identity and $a = a^*$, show that $\exp (ia) = u$ is unitary. Is the converse true?

If $a \in \operatorname {Re} \mathcal{A}$, show that $\left|a\right| \equiv (a^2)^{1/2} = a_+ + a_-$, where $a_+$ and $a_-$ are the positive and negative parts of $a$ (that is, the unique positive elements with $a = a_+ - a_-$ and $a_+ a_- = a_- a_+ = 0$).

Let $N$ be a normal operator on $\mathcal{H}$ with spectral measure $E$. If $\Delta$ is a clopen subset of $\sigma (N)$, show that $E(\Delta )$ is the Riesz idempotent associated with $\Delta$.

Prove Theorem II.7.6 and its corollaries by using the Spectral Theorem.

Obtain Theorem II.7.11 as a consequence of the following result (Theorem 2.3). If $N$ is a normal operator on $\mathcal{H}$ with spectral measure $E$ and $B(\sigma (N))$ is the $C^*$-algebra of bounded Borel functions on $\sigma (N)$, then the map $\phi \mapsto \phi (N)$ is a representation of the $C^*$-algebra $B(\sigma (N))$. If $\{ \phi_i\}$ is a net in $B(\sigma (N))$ such that $\int \phi_i \, d\mu \to 0$ for every $\mu$ in $M(\sigma (N))$, then $\phi_i(N) \to 0$ (WOT). This map is unique in the sense that if $\tau : B(\sigma (N)) \to \mathcal{B}(\mathcal{H})$ is a representation such that $\tau (z) = N$ and $\tau (\phi_i) \to 0$ (WOT) whenever $\{ \phi_i\}$ is a bounded net in $B(\sigma (N))$ such that $\int \phi_i \, d\mu \to 0$ for every $\mu$ in $M(\sigma (N))$, then $\tau (\phi ) = \phi (N)$ for all $\phi$ in $B(\sigma (N))$.

If $A$ is self-adjoint, prove that $\exp (iA)$ is unitary.

If $A \in \mathcal{B}(\mathcal{H})$ and $f \in \mathrm{Hol}(A)$, then $f(\sigma_p(A)) \subseteq \sigma_p(f(A))$. If $f$ is not constant on any component of its domain, then $f(\sigma_p(A)) = \sigma_p(f(A))$.

If $A \in \mathcal{B}(\mathcal{H})$ and $f \in \mathrm{Hol}(A)$, then $f(\sigma_{ap}(A)) = \sigma_{ap}(f(A))$.

Let $A \in \mathcal{F}$ and suppose $f$ is analytic in a neighborhood of $\sigma (A)$ and does not vanish on $\sigma_e(A)$. Show that $f(A) \in \mathcal{F}$ and find $\operatorname {ind} f(A)$.

Prove Proposition 6.9: if $\lambda$ is an isolated point of $\sigma (A)$, the following statements are equivalent.

1. $\lambda \notin \sigma_{le}(A) \cap \sigma_{re}(A)$.

2. The Riesz idempotent $E(\lambda )$ has finite rank.

3. $A - \lambda \in \mathcal{F}$ and $\operatorname {ind}(A - \lambda ) = 0$.

Suppose that the spectrum of some $x \in A$ is not connected. Prove that $A$ contains a nontrivial idempotent 2. (This is defined in Exercise 2).

Prove also that $A = A_{0} \oplus A_{1}$, where

Suppose $x \in A, x_{n} \in A$, and $\lim x_{n} = x$. Suppose $\Omega$ is an open set in $\mathscr {C}$ that contains a component of $\sigma (x)$. Prove that $\sigma \left(x_{n}\right)$ intersects $\Omega$ for all sufficiently large $n$. (This strengthens Theorem 10.20). Hint: If $\sigma (x) \subset \Omega \cup \Omega_{0}$, where $\Omega_{0}$ is an open set disjoint from $\Omega$, consider the function $f$ that is 1 in $\Omega , 0$ in $\Omega_{0}$.

Suppose $A$ is a commutative Banach algebra, $x \in A$, and $f$ is holomorphic in some open set $\Omega \subset \mathcal{C}$ that contains the range of $\hat{x}$. Prove that there exists $y \in A$ such that $\hat{y} = f \circ \hat{x}$, that is, such that $h(y) = f(h(x))$ for every complex homomorphism $h$ of $A$. Prove that $y$ is uniquely determined by $x$ and $f$ if $A$ is semisimple.

Prove that every normal invertible $T \in \mathscr {B}(H)$ is the exponential of some normal $S \in \mathscr {B}(H)$.