Polar decomposition

A polar decomposition of a linear transformation on a finite-dimensional Euclidean (or unitary) space 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 on has a polar decomposition

where is a positive semi-definite self-adjoint linear transformation and is an orthogonal (or unitary) linear transformation; moreover, is uniquely defined. If is non-degenerate, then is even positive definite and is also uniquely defined. A polar decomposition on a one-dimensional unitary space coincides with the trigonometric representation of a complex number as .

A.L. Onishchik

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

where is a partial isometric operator and is a positive operator. Any closed operator has a polar decomposition, moreover, (which is often denoted by ), and maps the closure of the domain of the self-adjoint operator into the closure of the range of (the von Neumann theorem, see ). A polar decomposition becomes unique if the source and target subspaces of are required to coincide with and , respectively. On the other hand, can be always chosen unitary, isometric or co-isometric, depending on the relation between the codimensions of the subspaces and . In particular, if

then can be chosen unitary and there is a Hermitian operator such that . Then the polar decomposition of takes the form

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 is a representation of a normal functional on in the form , where is a positive normal functional on , is a partial isometry (i.e. and are projectors), and multiplication is understood as the action on of the operator which is adjoint to left multiplication by in : for all . A polar decomposition can always be realized so that the condition is fulfilled. Under this condition a polar decomposition is unique.

Any bounded linear functional on an arbitrary -algebra can be considered as a normal functional on the universal enveloping von Neumann algebra ; the corresponding polar decomposition is called the enveloping polar decomposition of the functional . The restriction of the functional to is called the absolute value of and is denoted by ; the following properties determine the functional uniquely:

In the case when 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 -algebras to studies of positive functionals. It enables one, for example, to construct for each a representation of the algebra on which has a vector realization (i.e. there are vectors in such that , ). The representation constructed from the positive functional using the GNS-construction (of Gel'fand–Naimark–Segal) has that property.

The polar decomposition of an element of a -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 on a Hilbert space the positive term belongs to the -algebra generated by , but for the partial isometric term one can only state that it belongs to the von Neumann algebra generated by . That is why one defines and uses the so-called enveloping polar decomposition of an element : , where and is a partial isometric element in the universal enveloping von Neumann algebra (it is assumed that is canonically imbedded in ).

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)