The generalized polar decomposition, the weak complementarity and the parallel sum for adjointable operators on Hilbert $C^*$-modules

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.