  The effect of matrix perturbations on the polar decomposition has been
studied by several authors and various results are known. However, for
operators between infinite-dimensional spaces the problem has not been
considered so far. Here, we prove in particular that the partial isometry in
the polar decomposition of an operator is stable under perturbations, given
that kernel and range of original and perturbed operator satisfy a certain
condition. In the matrix case, this condition is weaker than the usually
imposed equal-rank condition. It includes the case of semi-Fredholm operators
with agreeing nullities and deficiencies, respectively. In addition, we prove a
similar perturbation result where the ranges or the kernels of the two
operators are assumed to be sufficiently close to each other in the gap metric.
