REPRESENTATION AND APPROXIMATION OF THE POLAR FACTOR OF AN OPERATOR ON A HILBERT SPACE

Let H be a complex Hilbert space and let B(H) be the algebra of all bounded linear operators on H. The polar decomposition theorem asserts that every operator T is an element of B(H) can be written as the product T = V P of a partial isometry V is an element of B(H) and a positive operator P is an element of B(H) such that the kernels of V and P coincide. Then this decomposition is unique. V is called the polar factor of T. Moreover, we have automatically P = vertical bar T vertical bar = (T*T) 1/2. Unlike P, we have no representation formula that is required for V. In this paper, we introduce, for T is an element of B(H), a family of functions called a "polar function" for T, such that the polar factor of T is obtained as a limit of a net built via continuous functional calculus from this family of functions. We derive several explicit formulas representing different polar factors. These formulas allow new for methods of approximations of the polar factor of T.