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

References

[a1]	I.C. [I.Ts. Gokhberg] Gohberg, M.G. Krein, "Introduction to the theory of linear nonselfadjoint operators" , Transl. Math. Monogr. , 18 , Amer. Math. Soc. (1969) (Translated from Russian)