Spectral theorem in terms of the multiplication operator

Under what conditions on the complex numbers $\left\{ \lambda_{k}\right\}_{k = 1}^{n}$ does the diagonal operator $A\left(x_{1}, \ldots , x_{n}\right) = \left(\lambda_{1} x_{1}, \ldots , \lambda_{n} x_{n}\right)$ in $l_{2}(n)$ have a cyclic vector? State the condition for the existence of a cyclic vector in terms of the characteristic polynomial of an arbitrary operator $A: \mathbb {C}^{n} \rightarrow \mathbb {C}^{n}$.

Under what conditions on the bounded sequence of complex numbers $\left\{ \lambda_{k}\right\}_{k = 1}^{\infty }$ does the diagonal operator

in $l_{2}$ have a cyclic vector?

Prove that the operator $(A x)(t) = a(t) x(t)$ of multiplication by a continuous strictly monotone function $a(t)$ has a cyclic vector:

Prove that the operator $(A x)(t) = a(t) x(t)$ of multiplication by a continuous non-monotone function $a(t)$, acting in the space $C[0,1]$, does not have a cyclic vector.

Let $A$ be a bounded self-adjoint operator with simple spectrum in a Hilbert space $H$. Prove that all its eigenvalues (if any) have multiplicity 1.

Prove that a compact self-adjoint operator in a Hilbert space possesses a cyclic vector precisely when all its eigenvalues are simple.

Let $A$ be an operator with a cyclic vector in an infinite-dimensional Hilbert space $H$. Prove that the operators $I, A, A^{2}, \ldots$ are linearly independent.

Prove that the operator $T_{l}$ of left shift in the space $l_{2}$ has a "supercyclic" vector, i.e., a vector $x$ such that the set $\left\{ \frac{T_{l}^{n} x}{\left\| T_{l}^{n} x\right\| }\right\}_{n = 0}^{\infty }$ is dense on the unit sphere. Does there exist an invertible operator with a supercyclic vector?

Let $A$ be a self-adjoint operator in a separable Hilbert space $H$. Prove that there exists a decomposition into a direct sum $H = \bigoplus_{n = 1}^{N} H_{n}$, where $N = 1,2, \ldots$ or $\infty$, such that $A H_{n} \subset H_{n}$ and for every $n = 1,2, \ldots , N$ there exists a vector $x_{n} \in H_{n}$ that is cyclic for the restriction of $A$ to $H_{n}$, i.e., $\overline{\operatorname {Lin}\left\{ A^{k} x_{n}\right\}_{k = 0}^{\infty }} = H_{n}$.

Find, in the space $L_{2}[-1,1]$, the resolution of the identity of the operator of multiplication:

For the operator of multiplication from the previous problem:

Give an example of two bounded self-adjoint operators $A$ and $B$ in the space $l_{2}$ whose spectra coincide, with $\sigma_{\mathrm{p}}(A) = \sigma_{\mathrm{p}}(B)$ (and the multiplicities of the corresponding eigenvalues coincide), $\sigma_{\mathrm{c}}(A) = \sigma_{\mathrm{c}}(B)$, but the operators themselves are not unitarily equivalent.

Find the resolution of the identity of the operator of multiplication by $t^{3}$ in the space $L_{2}[0,1]$. Reduce the operator to the form of multiplication by the independent variable (find an isometric isomorphism and measure $\mu$).

Find the resolution of the identity of the operator of multiplication by $\sin t$ in the space $L_{2}[0, \pi / 2]$. Reduce the operator to the form of multiplication by the independent variable (find an isometric isomorphism and measure $\mu$).

Find the resolution of the identity of the operator of multiplication by $\sin t$ in the space $L_{2}(\mathbb {R})$.

For the operator of problem 13.77, find a unitary isomorphism

such that the operator $U A U^{-1}$, restricted to the space $L_{2}\left([a, b], \mu_{i}\right)$, is the operator of multiplication by the independent variable. Find $a, b$ and the measures $\mu_{i}$.

Reduce the operator

acting in the space $L_{2}[0,1]$, to the form of multiplication by the independent variable (find an isometric isomorphism and measure $\mu$).

Let $A$ be a linear self-adjoint operator in a finite-dimensional space. Find a unitary operator $U$, a measure $\mu$, and a function $a$ such that $U A U^{-1} = B$, where $B \in \mathcal{B}\left(L_{2}([0,1], \mu )\right),(B x)(t) = a(t) x(t)$.

In the space $l_{2}(\mathbb {Z})$, consider the operator $A = T_{r}+T_{l}$, where $T_{r}, T_{l}$ are the operators of right and left shift, respectively. Find a unitary operator $U$ and a function $a$ such that $U A U^{-1} = B$, where $B \in \mathcal{B}\left(L_{2}[-\pi , \pi ]\right), B x(t) = a(t) x(t)$.

Prove that if an operator $A \in \mathcal{B}(H)$ in a Hilbert space is unitarily equivalent to an operator $B \in \mathcal{B}\left(L_{2}[0,1]\right)$ of multiplication by a function $B x(t) = a(t) x(t)$, then $A$ is a normal operator.

Prove that every normal operator $A$ in a Hilbert space $H$ is unitarily equivalent to an operator of multiplication by some function $a$ in the space $L_{2}\left([0,1]^{2}, \mu \right)$ with some measure $\mu$.

Prove that the operators

acting in the space $L_{2}[0,1]$ (here $L K(t)$ is the Cantor function, see the definition on p. 72), are not unitarily equivalent.

Let $\mu$ and $v$ be two nonnegative Borel measures on the segment $[0,1]$, and let $T_{\mu }$ and $T_{v}$ be the operators of multiplication $x(t) \mapsto t x(t)$, acting in the spaces $L_{2}([0,1], \mu )$ and $L_{2}([0,1], v)$ respectively. These operators are unitarily equivalent if and only if the measures $\mu$ and $v$ are equivalent. ¹

Using Theorem 13.8, prove Theorem 13.9.

Theorem 13.8: For every self-adjoint operator $A$ in a separable Hilbert space $H$ there exist a decomposition $H = \bigcup_{k \in \mathcal{K}} H_{k}$, where $\mathcal{K} = \mathbb {N} \cup \left\{ \infty \right\}$ (some of the subspaces may be zero), a family of pairwise singular Borel measures $\left\{ \mu_{k}\right\}_{k \in \mathcal{K}}$ with supports $\operatorname {supp} \mu_{k} =: X_{k} \subset \sigma (A)$ (some of the sets $X_{k}$ may be empty), and a unitary isomorphism

such that the restriction of $A$ to $H_{k}$ is unitarily equivalent to the orthogonal direct sum of $k$ copies of the operator of multiplication by the independent variable in $L_{2}\left(X_{k}, \mu_{k}\right)$:

where $M_{k} \in \mathcal{B}\left(L_{2}\left(X_{k}, \mu_{k}\right)\right)$, $\left(M_{k} x\right)(t) = t x(t)$.

Theorem 13.9: Two self-adjoint operators $A$ and $B$ in a Hilbert space are unitarily equivalent if and only if their multiplicity functions coincide and the spectral measures defined in Theorem 13.8 are equivalent: $\mu_{k, A} \sim \mu_{k, B}$ for every $k \in \mathcal{K}$. (Here the multiplicity function of $A$ is $n_{A}(\lambda ) = 0$ outside $\sigma (A)$, $n_{A}(\lambda ) = k$ for $\lambda \in X_{k}$, $k \in \mathcal{K}$.)

Determine whether among the following operators there are unitarily equivalent ones: $A_{1} \in \mathcal{B}\left(L_{2}[0,1]\right),\left(A_{1} x\right)(t) = t x(t) ;$ $A_{2} \in \mathcal{B}\left(L_{2}[-1,1]\right),\left(A_{2} x\right)(t) = \left|t\right| x(t) ;$ $A_{3} \in \mathcal{B}\left(L_{2}[0,1]\right),\left(A_{3} x\right)(t) = t^{2} x(t) ;$ $A_{4} \in \mathcal{B}\left(L_{2}[0,1]\right),\left(A_{4} x\right)(t) = t^{3} x(t) ;$ $A_{5} \in \mathcal{B}\left(L_{2}[0,1]\right),\left(A_{5} x\right)(t) = \sqrt{t} x(t) ;$ $A_{6} \in \mathcal{B}\left(L_{2}[0, \pi / 2]\right),\left(A_{6} x\right)(t) = \sin t x(t) ;$ $A_{7} \in \mathcal{B}\left(L_{2}[0, \pi ]\right),\left(A_{7} x\right)(t) = \sin t x(t) ;$ $A_{8} \in \mathcal{B}\left(L_{2}[0,1]\right),\left(A_{8} x\right)(t) = (1-t) x(t) ;$ $A_{9} \in \mathcal{B}\left(L_{2}[0,1]\right),\left(A_{9} x\right)(t) = L K(t) x(t)$.

Construct the canonical functional model:

Construct the canonical functional model for the following operators in the space $l_{2}$: