Characterizing and testing partial isometries

Suppose $(X, \mathcal{S}, \mu )$ is a $\sigma$-finite measure space and $h \in L^{\infty }(\mu )$. As usual, let $M_{h} \in \mathcal{B}\left(L^{2}(\mu )\right)$ denote the multiplication operator defined by $M_{h} f = f h$. Prove that $M_{h}$ is a partial isometry if and only if there exists a set $E \in \mathcal{S}$ such that $\left|h\right| = \chi_{E}$.

Suppose $T$ is an isometry on a Hilbert space. Prove that $T^{*}$ is a partial isometry.

Suppose $T$ is a bounded operator on a Hilbert space $V$. Prove that $T$ is a partial isometry if and only if $T^{*} T = P_{U}$ for some closed subspace $U$ of $V$.

Let $H$ be a Hilbert space and operator $A \in \mathcal{L}(H)$ be continuously invertible. Prove that

Let $H$ be a Hilbert space. An operator $U \in \mathcal{L}(H)$ is called partially isometric if $\left\| U x\right\| = \left\| x\right\| , x \in (\operatorname {Ker} U)^{\perp }$.

Let $H$ be a Hilbert space, $A \in \mathcal{L}(H)$. Representation $A = U B$, where operator $B \in \mathcal{L}(H)$ is nonnegative, operator $U \in \mathcal{L}(H)$ is partially isometric and $\operatorname {Ker} U = \operatorname {Ker} B$, is called the polar decomposition of $A$ (compare with problem 9.64). Prove that the polar decomposition exists and is unique, moreover $B = \left|A\right|$.

Let $A = U\left|A\right|$ be the polar decomposition of an operator $A \in \mathcal{L}(H)$. Prove that $\left|A^{*}\right| = U\left|A\right| U^{*}, A^{*} = U^{*}\left|A^{*}\right|, A = \left|A^{*}\right| U, A^{*} = \left|A\right| U^{*}$.

Find the polar decomposition of an operator $A \in \mathcal{L}\left(l_{2}\right)$.

Let $A = U\left|A\right|$ be the polar decomposition of an operator $A \in \mathcal{L}(H)$.

Let $H_{1}$ and $H_{2}$ be Hilbert spaces, $W \in \mathcal{B}\left(H_{1}, H_{2}\right)$. Prove that $W$ is a partially isometric operator if and only if $W W^{*} W = W$. How does the operator $W^{*}$ act?

The Hilbert transform of a function $x \in L_{2}[0,2 \pi ]$ is the function $y(t) = \frac{1}{2 \pi } \int_{0}^{2 \pi } x(s) \operatorname {ctg} \frac{t-s}{2} d s.{ }$¹ Prove that the operator $A: x \mapsto y$ is a partial isometry.

Under what conditions on the sequence $\left\{ \lambda_{n}\right\}_{1}^{\infty } \in l_{\infty }$ is the operator $A x = \left(\lambda_{1} x_{1}, \lambda_{2} x_{2}, \lambda_{3} x_{3}, \ldots \right)$ a partial isometry in the space $l_{2}$? Under what conditions on the function $a \in L_{\infty }[0,1]$ is the operator $(A x)(t) = a(t) x(t)$ a partial isometry in the space $L_{2}[0,1]$?

Let $H_{1}$ and $H_{2}$ be Hilbert spaces, and let an operator $A \in \mathcal{B}\left(H_{1}, H_{2}\right)$. Prove that there exists a partially isometric operator $W: H_{1} \rightarrow H_{2}$ and nonnegative operators $S \in \mathcal{B}\left(H_{1}\right)$ and $R \in \mathcal{B}\left(H_{2}\right)$ such that $A = W S = R W$. Here the pair of operators $W, S$ is uniquely determined by the condition $\operatorname {Ker} W = \operatorname {Ker} S$, and the pair $W, R$ by the condition $(\operatorname {Im} W)^{\perp } = \operatorname {Ker} R$.

Let $W \in \mathcal{B}(\mathcal{H})$ and show that the following statements are equivalent: (a) $W$ is a partial isometry (that is, an operator such that for $h$ in $(\ker W)^\perp$, $\left\| Wh\right\| = \left\| h\right\|$); (b) $W^*$ is a partial isometry; (c) $W^*W$ is a projection; (d) $WW^*$ is a projection; (e) $WW^*W = W$; (f) $W^*WW^* = W^*$.

If $W$ is a partial isometry, show that $W^*W$ is the projection onto the initial space of $W$ (that is, the space $(\ker W)^\perp$) and $WW^*$ is the projection onto the final space of $W$ (that is, the space $\operatorname {ran} W$).

If $W_1, W_2$ are partial isometries, define $W_1 \precsim W_2$ to mean that $W_1^* W_1 \leq W_2^* W_2$, $W_1 W_1^* \leq W_2 W_2^*$, and $W_2 h = W_1 h$ whenever $h$ is in the initial space of $W_1$. Show that $\precsim$ is a partial ordering on the set of partial isometries and that a partial isometry $W$ is a maximal element in this ordering if and only if either $W$ or $W^*$ is an isometry.

Using the terminology of the preceding exercise, show that the extreme points of ball $\mathcal{B}(\mathcal{H})$ are the maximal partial isometries. (See Exercise V.7.10.)

If $U$ is a partial isometry, show that the following statements are equivalent:

1. $\ker (1 - U) = (0)$;

2. $\ker (1 - U^*) = (0)$;

3. $\operatorname {ran}(1 - U)$ is dense;

4. $\operatorname {ran}(1 - U^*)$ is dense.

Let $U$ be a partial isometry with initial and final spaces $\mathcal{M}$ and $\mathcal{N}$, respectively. Show that the following statements are equivalent:

1. $(1 - U)\mathcal{M}$ is dense;

2. $(1 - U^*)\mathcal{N}$ is dense;

3. $\ker (U^* - U^*U) = (0)$;

4. $\ker (U - UU^*) = (0)$.

Find a partial isometry $U$ such that $\ker (1 - U) = (0)$ but $(1 - U)(\ker U)^{\perp }$ is not dense.

Find the components of the set of partial isometries.