Functional calculus of a bounded operator

(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 $A = \exp (B)$?