  This paper deals mainly with some aspects of the adjointable operators on
Hilbert $C^*$-modules. A new tool called the generalized polar decomposition
for each adjointable operator is introduced and clarified. As an application,
the general theory of the weakly complementable operators is set up in the
framework of Hilbert $C^*$-modules. It is proved that there exists an operator
equation which has a unique solution, whereas this unique solution fails to be
the reduced solution. Some investigations are also carried out in the Hilbert
space case. It is proved that there exist a closed subspace $M$ of certain
Hilbert space $K$ and an operator $T\in \mathbb{B}(K)$ such that $T$ is
$(M,M)$-weakly complementable, whereas $T$ fails to be $(M,M)$-complementable.
The solvability of the equation $$A:B=X^*AX+(I-X)^*B(I-X) \quad
(X\in\mathbb{B}(H))$$ is also dealt with in the Hilbert space case, where
$A,B\in \mathbb{B}(H)$ are two general positive operators, and $A:B$ denotes
their parallel sum. Among other things, it is shown that there exist certain
positive operators $A$ and $B$ on the Hilbert space $\ell^2(\mathbb{N})\oplus
\ell^2(\mathbb{N})$ such that the above equation has no solution.
