Let $mathbb{C}^{ntimes n}$ be the set of all $n times n$ complex matrices. For any Hermitian positive semi-definite matrices $A$ and $B$ in $mathbb{C}^{ntimes n}$, their new common upper bound less than $A+B-A:B$ is constructed, where $(A+B)^dag$ den
otes the Moore-Penrose inverse of $A+B$, and $A:B=A(A+B)^dag B$ is the parallel sum of $A$ and $B$. A factorization formula for $(A+X):(B+Y)-A:B-X:Y$ is derived, where $X,Yinmathbb{C}^{ntimes n}$ are any Hermitian positive semi-definite perturbations of $A$ and $B$, respectively. Based on the derived factorization formula and the constructed common upper bound of $X$ and $Y$, some new and sharp norm upper bounds of $(A+X):(B+Y)-A:B$ are provided. Numerical examples are also provided to illustrate the sharpness of the obtained norm upper bounds.
The parallel sum for adjoinable operators on Hilbert $C^*$-modules is introduced and studied. Some results known for matrices and bounded linear operators on Hilbert spaces are generalized to the case of adjointable operators on Hilbert $C^*$-modules
. It is shown that there exist a Hilbert $C^*$-module $H$ and two positive operators $A, Binmathcal{L}(H)$ such that the operator equation $A^{1/2}=(A+B)^{1/2}X, Xin cal{L}(H)$ has no solution, where $mathcal{L}(H)$ denotes the set of all adjointable operators on $H$.
For two positive definite adjointable operators $M$ and $N$, and an adjointable operator $A$ acting on a Hilbert $C^*$-module, some properties of the weighted Moore-Penrose inverse $A^dag_{MN}$ are established. When $A=(A_{ij})$ is $1times 2$ or $2ti
mes 2$ partitioned, general representations for $A^dag_{MN}$ in terms of the individual blocks $A_{ij}$ are studied. In case $A$ is $1times 2$ partitioned, a unified representation for $A^dag_{MN}$ is presented. In the $2times 2$ partitioned case, an approach to constructing Moore-Penrose inverses from the non-weighted case to the weighted case is provided. Some results known for matrices are generalized in the general setting of Hilbert $C^*$-module operators.
