Commutants of operators

Let $H$ be a Hilbert space over the field $\mathbb {K}$, and $A \in \mathcal{L}(H)$ be such that $A B = B A$ for all $B \in \mathcal{L}(H)$. Prove that $A = \gamma I$ for some $\gamma \in \mathbb {K}$.

If $A \in \mathcal{B}(\mathcal{H})$ and $AT=TA$ for every compact operator $T$, show that $A$ is a multiple of the identity operator.

Suppose $T$ is a compact normal operator on a $\mathbb {C}$-Hilbert space such that $\dim \ker (T-\lambda ) \leq 1$ for all $\lambda$ in $\mathbb {C}$. Show that if $A \in \mathcal{B}(\mathcal{H})$ and $AT=TA$, then $A = \phi (T)$ for some $\phi$ in $l^{\infty }(\mathbb {C})$.

Definition 7.10 (context): Denote by $l^{\infty }(\mathbb {C})$ all the bounded functions $\phi : \mathbb {C} \rightarrow \mathbb {C}$. If $T$ is a compact normal operator satisfying $T = \sum_{n=1}^{\infty } \lambda_{n} P_{n}$ (7.7), define $\phi (T): \mathcal{H} \rightarrow \mathcal{H}$ by

where $P_{0}$ is the projection of $\mathcal{H}$ onto $\ker T$.

Prove a converse to Exercise 8: if $T$ is a compact normal operator such that $\left\{ A \in \mathcal{B}(\mathcal{H}): AT=TA\right\} = \left\{ \phi (T): \phi \in l^{\infty }(\mathbb {C})\right\}$, then $\dim \ker (T-\lambda ) \leq 1$ for all $\lambda$ in $\mathbb {C}$.

Let $T$ be a compact normal operator and show that $\dim \ker (T-\lambda ) \leq 1$ for all $\lambda$ in $\mathbb {C}$ if and only if there is a vector $h$ in $\mathcal{H}$ such that $\left\{ p(T)h: p \text{ is a polynomial in one variable}\right\}$ is dense in $\mathcal{H}$. (Such a vector $h$ is called a cyclic vector for $T$.)

If $\lambda \in \mathbb {C}$, let $\delta_{\lambda }$ be the unit point mass at $\lambda$; that is, $\delta_{\lambda }$ is the measure on $\mathbb {C}$ such that $\delta_{\lambda }(\Delta ) = 1$ if $\lambda \in \Delta$ and $\delta_{\lambda }(\Delta ) = 0$ if $\lambda \notin \Delta$. If $\left\{ \lambda_{1}, \lambda_{2}, \ldots \right\}$ is a bounded sequence of distinct complex numbers and $\left\{ \alpha_{n}\right\}$ is a sequence of real numbers with $\alpha_{n} > 0$ and $\sum_{n} \alpha_{n} < \infty$, let $\mu = \sum_{n=1}^{\infty } \alpha_{n} \delta_{\lambda_{n}}$; so $\mu$ is a finite measure. If $\phi \in l^{\infty }(\mathbb {C})$, let $M_{\phi }$ be the multiplication operator on $L^{2}(\mu )$. Define $T: L^{2}(\mu ) \rightarrow L^{2}(\mu )$ by $(Tf)(\lambda_{n}) = \lambda_{n} f(\lambda_{n})$. Prove:

The commutant of the bilateral shift is the set of all multiplications.

The commutant of the unilateral shift is the set of all restrictions to $\mathbf{H}^2$ of multiplications by multipliers in $\mathbf{H}^{\infty }$.

Every operator that commutes with the unilateral shift is the limit (strong operator topology) of a sequence of polynomials in the unilateral shift.

Prove that every continuous operator in $S(\mathbb {R}^{n})$ that commutes with the operators $M_k$, $1\leq k\leq n$, (see Problem 580) is the operator of multiplication by a function.

Prove that every continuous operator in $S(\mathbb {R}^{n})$ that commutes with the operators $M_k$ and $D_k$, $1\leq k\leq n$, (see Problem 580) is a scalar operator.

Let $S$ be some set of operators in a linear space $L$. Let $S'$ be the collection of operators in $L$ that commute with all the operators in $S$. For what operators $A$ does the equality $\mathfrak A(A)'=\mathfrak A(A)$ hold?

In the space of $2n\times 2n$ matrices find a subspace of dimension $1+n^2$ consisting of pairwise commuting matrices.

Let $A$ be the operator on $L^{2}\left({[(0,\infty ),dx]}\right)$ given by the formula $Af(x)=\int_0^\infty [f(y)/(x+y)]\, dy$. Prove that $A$ commutes with the dilation operators $L(a):f(x)\mapsto f(ax)$.

Suppose that $B$ and $C$ are bounded selfadjoint operators that commute. Prove that there exists a bounded selfadjoint operator $A$ such that $B$ and $C$ are functions of $A$.

Let $A$ be a selfadjoint operator with simple spectrum. Prove that every bounded operator $B$ that commutes with $A$ is a function of $A$.

Prove that any family of pairwise commuting bounded selfadjoint operators on a separable Hilbert space can be simultaneously reduced to the form of multiplication by a function.