Higham's approximations of polar factors of operators

Abstract

Let H be a complex Hilbert space and $B\left(H\right)$ be the algebra of all bounded linear operators on H. For any operator $A\in B\left(H\right)$, let $\left|A\right|:={\left(A^{⁎}A\right)}^{\frac{1}{2}}$ and $A^{†}$ be the modulus and the Moore-Penrose inverse of A, respectively. The polar factor of an operator $A\in B\left(H\right)$ is, by the polar decomposition theorem, the unique partial isometry $V_A\in B\left(H\right)$ whose kernel coincides with that of A such that $A=V_A\left|A\right|$. In this paper, we show that if $A\in B\left(H\right)$ has closed range then the sequence$X_0=A,X_{n+1}=\frac{1}{2}(X_n+X_n^{⁎†}),(n\ge 0),$ converges to $V_A$ in the norm topology of $B\left(H\right)$. This extends Higham's matrix result to the setting of operators with closed range. Furthermore, we obtain another approximation of the polar factors of operators in $B\left(H\right)$ and discuss a related linear preserving problem.