# Lecture 5: Polar Decomposition, Compact Operators
###### tags: `224a`
$\renewcommand{\ker}{\mathrm{Ker}}$
$\newcommand{\ran}{\mathrm{Ran}}$
## Range and Kernel
Let $\ker(A)$ and $\ran(A)$ denote the kernel and range of $A\in L(H)$. These are related as:
$$ \ran(A^*)^\perp = \ker(A),\qquad \overline{\ran(A^*)}=\ker(A)^\perp,$$
where the second equality can be seen by taking the orthogonal complement of the first one and noting that $(M^\perp)^\perp =\overline{M}$ for any subspace $M$ (homework).

While $\ker(A)$ is always a closed subspace, this is not true of $\ran(A)$, and this distinction is important for instance because the projection theorem, orthogonal decomposition, and existence of orthonormal bases only work for *closed* subspaces.

**Example.** To see all of this in action, consider the Volterra operator
$$(Vf)(x) = \int_0^x f(y)dy$$
on $L^2(0,1)$ from the last lecture, which we showed was bounded. Its adjoint can be calculated from the definition by observing
\begin{align*}
(g,Vf) &= \int_0^1 \overline{g(x)} (\int_0^1 \{y\le x\} f(y)dy) dx\\
&= \int_0^1 (\int_0^1 \{y\le x\} \overline{g(x)} dx) f(y)dy\quad\textrm{by Fubini}
\\&=(V^*g,f)\end{align*}
by definition, so we must have
$$ (V^*f)(x)=\int_x^1 f(y)dy.$$

It is now evident that $V$ is not self-adjoint, since for any strictly positive function $f$, $Vf$ is strictly increasing whereas $V^*f$ is strictly decreasing.

To compute $\ker(V)$, observe that by the fundamental theorem of calculus (or more precisely its Lebesgue version, https://en.wikipedia.org/wiki/Lebesgue_differentiation_theorem), $(Vf)(x)$ is an almost everywhere differentiable function of $f$, with
$$\frac{d}{dx}(Vf)(x)=f(x).$$
Thus if $Vf=0$ a.e. we must have $f=0$ a.e., so $\ker(V)=\{0\}$.

The above calculation also reveals that $\ran(V)\neq L^2(0,1)$ (since there are continuous functions in $L^2$ which are not differentiable a.e.), yielding an example of a range which is not closed. However, we do know that
$$\overline{\ran(V)} = \ker(V^*)^\perp = L^2(0,1)$$
since $\ker(V^*)=0$ as well.

**Remark 1.** An operator of type $$(T_Kf)(x):=\int_0^1 K(x,y)f(y)dy$$ is called an *integral kernel* operator, and the Fubini argument above shows that whenever such an operator is bounded we have
$$ (T_K)^* = T_{K^*},$$
where $K^*(x,y)=\overline{K(y,x)}$.

## Projection and Unitary Operators

A *projection* is an operator which satisfies $P^2=P$, and an *orthogonal projection* further satisfies $P=P^*$. It is easy to check that $\ran(P)$ is closed for a projection, and there is a 1-1 correspondence between orthogonal projections and closed subspaces of $H$ by $H=\ran(P)\oplus \ker(P)$.

A *partial isometry* is an operator $U:H\rightarrow H$ satisfying
$$ \|Ux\|=\|x\|\quad\forall x\in \ker(A)^\perp,$$
which can be seen to be equivalent to
$$ (Ux,Uy)=(x,y)\quad\forall x,y\in\ker(A)^\perp$$
by the polarization identity. Such an operator is called *unitary* if $\ker(U)=\{0\}$, in which case $U$ is a bijection.

It is easy to verify that $U^*U=P_{\ker(U)^\perp}$ and $UU^* = P_{\ran(U)}$. The first fact implies that the range of a unitary is always closed since a sequence $Ux_n$ is Cauchy if and only if $x_n$ is Cauchy.

Intuitively, we think of unitaries as "rotations" and of partial isometries as "geometry preserving" embeddings of one subspace into another.

**Example.** The right shift operator on $\ell^2(\mathbb{N})$ is a partial isometry with $\ker=\{e_1\}$ whereas the right shift operator on $\ell^2(\mathbb{Z})$ is a unitary.

## The Polar Decomposition
The polar decomposition allows one to express any bounded operator as a product of a partial isometry and a positive operator. Let $|A|=\sqrt{A^*A}$ denote the absolute value of $A$.
**Theorem 1.** If $A\in L(H)$ then there is a unique partial isometry $U$ such that
$$A=U|A|$$
with $\ker(U)=\ker(A)$.

The proof is given in Reed and Simon Theorem VI.10.

Suppose we could prove that every self-adjoint operator (such as $|A|$) can be "diagonalized": $|A|=VDV^*$ for some "diagonal" $D$ (we will prove this soon, though the definition of diagonal is different from the finite case). Then the polar decomposition gives:
$$ A = U(VDV^*)=WDV^*$$
for some partial isometries $W,V$, which is a "singular value decomposition" for elements of $L(H)$.

## Compact Operators

Rather than trying to understand all self-adjoint operators, we begin by focusing on the a simpler subclass. The simplest operators in $L(H)$ are the *finite rank* operators:
$$A = \sum_{n\le N} (\phi_n, \cdot)\psi_n$$
for some $\phi_n,\psi_n\in H$. These operators always have closed range (since finite dimensional subspaces are closed), and can be understood using the methods of linear algebra.

The compact operators are a larger class which inherit many of the nice properties of finite rank operators.

**Definition.** An operator $T\in L(H)$ is *compact* if for every bounded set $B\subset H$, the closure of $T(B)$ is compact.

Compactness may seem like a weird condition, but it comes in handy because it allows one to mimick finite dimensional arguments where one "optimizes" over the unit norm ball in $\mathbb{R}^n$ to produce e.g. an eigenvector. This is not directly possible in $H$ because the norm ball $B=\{x\in H:\|x\|\le 1\}$ is not compact: consider the infinitely many basis vectors $e_1,e_2,\ldots$, which have pairwise distance $\sqrt{2}$.

A more intuitive understanding of this class is provided by the following theorem, which we will prove next time.

**Theorem.** An operator $T\in L(H)$ is compact iff it is the norm limit of finite rank operators.

Note that the content of the above theorem is that it is a *norm* limit. It is easy to see that every bounded operator is a limit of finite rank operators in the strong topology (i.e., there are finite rank $T_n$ such that for every $x$, $\|T_nx-Tx\|\rightarrow 0$ where the rate of convergence may depend on $x$.)

The most obvious non-example of a compact operator is the identity, since it maps the unit ball to itself. A large class of examples is provided by integral kernel operators.

**Theorem.** If $K\in L^2(0,1)^2$ then $T_K$ is a compact operator.

In particular, the Volterra operator above is compact. We will prove this theorem in the next lecture.