  We study a mapping $\tau_G$ of the cone ${\mathbf B}^+({\mathcal H})$ of
bounded nonnegative self-adjoint operators in a complex Hilbert space
${\mathcal H}$ into itself. This mapping is defined as a strong limit of
iterates of the mapping ${\mathbf B}^+({\mathcal H})\ni
X\mapsto\mu_G(X)=X-X:G\in{\mathbf B}^+({\mathcal H})$, where $G\in{\mathbf
B}^+({\mathcal H})$ and $X:G$ is the parallel sum. We find explicit expressions
for $\tau_G$ and establish its properties. In particular, it is shown that
$\tau_G$ is sub-additive, homogeneous of degree one, and its image coincides
with set of its fixed points which is the subset of ${\mathbf B}^+({\mathcal
H})$, consisting of all $Y$ such that ${\rm ran\,} Y^{1/2}\cap{\rm ran\,}
G^{1/2}=\{0\}$. Relationships between $\tau_G$ and Lebesgue type decomposition
of nonnegative self-adjoint operator are established and applications to the
properties of unbounded self-adjoint operators with trivial intersections of
their domains are given.
