# Polar decomposition

A polar decomposition of a linear transformation on a finite-dimensional Euclidean (or unitary) space $L$ is a decomposition of the linear transformation into a product of a self-adjoint and an orthogonal (respectively, unitary) transformation (cf. Orthogonal transformation; Self-adjoint linear transformation; Unitary transformation). Any linear transformation $A$ on $L$ has a polar decomposition

$$A = S \cdot U ,$$

where $S$ is a positive semi-definite self-adjoint linear transformation and $U$ is an orthogonal (or unitary) linear transformation; moreover, $S$ is uniquely defined. If $A$ is non-degenerate, then $S$ is even positive definite and $U$ is also uniquely defined. A polar decomposition on a one-dimensional unitary space coincides with the trigonometric representation of a complex number $z$ as $z = re ^ {i \phi }$.

A.L. Onishchik

A polar decomposition of an operator $A$ acting on a Hilbert space is a representation of $A$ in the form

$$A = U T,$$

where $U$ is a partial isometric operator and $T$ is a positive operator. Any closed operator $A$ has a polar decomposition, moreover, $T = ( A ^ {*} A ) ^ {1/2}$( which is often denoted by $T = | A |$), and $U$ maps the closure $\overline{R}\; _ {A ^ {*} }$ of the domain of the self-adjoint operator $A$ into the closure $\overline{R}\; _ {A}$ of the range of $A$( the von Neumann theorem, see ). A polar decomposition becomes unique if the source and target subspaces of $U$ are required to coincide with $\overline{R}\; _ {A ^ {*} }$ and $\overline{R}\; _ {A}$, respectively. On the other hand, $U$ can be always chosen unitary, isometric or co-isometric, depending on the relation between the codimensions of the subspaces $\overline{R}\; _ {A ^ {*} }$ and $\overline{R}\; _ {A}$. In particular, if

$$\mathop{\rm dim} H \ominus \overline{R}\; _ {A ^ {*} } = \ \mathop{\rm dim} H \ominus \overline{R}\; _ {A} ,$$

then $U$ can be chosen unitary and there is a Hermitian operator $\Phi$ such that $U = \mathop{\rm exp} ( i \Phi )$. Then the polar decomposition of $A$ takes the form

$$A = \mathop{\rm exp} ( i \Phi ) | A | ,$$

entirely analogous to the polar decomposition of a complex number. Commutativity of the terms in a polar decomposition takes place if and only if the operator is normal (cf. Normal operator).

An expression analogous to the polar decomposition has been obtained for operators on a space with an indefinite metric (see , ).

A polar decomposition of a functional on a von Neumann algebra $A$ is a representation of a normal functional $f$ on $A$ in the form $f = u p$, where $p$ is a positive normal functional on $A$, $u \in A$ is a partial isometry (i.e. $u ^ {*} u$ and $u u ^ {*}$ are projectors), and multiplication is understood as the action on $p$ of the operator which is adjoint to left multiplication by $u$ in $A$: $f ( x) = p ( u x )$ for all $x \in A$. A polar decomposition can always be realized so that the condition $u ^ {*} f = p$ is fulfilled. Under this condition a polar decomposition is unique.

Any bounded linear functional $f$ on an arbitrary $C ^ {*}$- algebra $A$ can be considered as a normal functional on the universal enveloping von Neumann algebra $A ^ {\prime\prime}$; the corresponding polar decomposition $f = u p$ is called the enveloping polar decomposition of the functional $f$. The restriction of the functional $p$ to $A$ is called the absolute value of $f$ and is denoted by $| f |$; the following properties determine the functional $| f |$ uniquely:

$$\| | f | \| = \| f \| \ \textrm{ and } \ | f ( x) | ^ {2} \leq \| f \| \cdot | f | ( x ^ {*} x ) .$$

In the case when $A = C ( X)$ is the algebra of all continuous functions on a compactum, the absolute value of a functional corresponds to the total variation of the measure determined by it (cf. also Total variation of a function).

In many cases a polar decomposition of a functional allows one to reduce studies of functionals on $C ^ {*}$- algebras to studies of positive functionals. It enables one, for example, to construct for each $f \in A ^ \prime$ a representation $\pi$ of the algebra $A$ on which $f$ has a vector realization (i.e. there are vectors $\xi , \eta$ in $H _ \pi$ such that $f ( x) = ( \pi ( x) \xi , \eta )$, $x \in A$). The representation $\pi _ {| f | }$ constructed from the positive functional $| f |$ using the GNS-construction (of Gel'fand–Naimark–Segal) has that property.

The polar decomposition of an element of a $C ^ {*}$- algebra is a representation of the element as the product of a positive element and a partial isometric element. Polar decomposition is not valid for all elements: in the usual polar decomposition of an operator $T$ on a Hilbert space the positive term belongs to the $C ^ {*}$- algebra generated by $T$, but for the partial isometric term one can only state that it belongs to the von Neumann algebra generated by $T$. That is why one defines and uses the so-called enveloping polar decomposition of an element $a \in A$: $a = u t$, where $t = ( a ^ {*} a ) ^ {1/2} \in A$ and $u$ is a partial isometric element in the universal enveloping von Neumann algebra $A ^ {\prime\prime}$( it is assumed that $A$ is canonically imbedded in $A ^ {\prime\prime}$).

#### References

 [1] M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian) [2] J. Bognár, "Certain relations among the non-negativity properties of operators on spaces with an indefinite metric II" Stud. Scient. Math. Hung. , 1 : 1–2 (1966) pp. 97–102 (In Russian) [3] J. Dixmier, " algebras" , North-Holland (1977) (Translated from French)

V.S. Shul'man