Polar factor's representations and approximations

Let H be a complex Hilbert space and let $B\left(H\right)$ be the algebra of all bounded linear operators on H. The polar decomposition theorem asserts that every operator $T\in B\left(H\right)$ can be uniquely written as $T=V_T\left|T\right|$, where $\left|T\right|:={\left(T^{⁎}T\right)}^{\frac{1}{2}}$ and $V_T$ is a partial isometry, called the polar factor of T, whose kernel coincides with that of T. In this paper, we obtain various representations and approximations of the polar factors of operators.