In this paper, we state some results on product of operators with closed ranges and we solve the operator equation $TXS^*-SX^*T^*= A$ in the general setting of the adjointable operators between Hilbert $C^*$-modules, when $TS = 1$. Furthermore, by using some block operator matrix techniques, we nd explicit solution of the operator equation $TXS^*-SX^*T^*= A$.

In this paper, we study the concept of controlled ∗ -operator frames for space all adjointable operators on a Hilbert id="M4"> C -module H. Also, discuss characterizations id="M5"> and give some properties. Some illustrative examples are provided to advocate usability our results.

Suppose that $${\mathscr {E}}$$ and {F}}$$ are Hilbert $$C^*$$ -modules. We present a power-norm $$\left( \left\| \cdot \right\| ^{{\mathscr {E}}}_n:n\in {\mathbb {N}}\right) $$ based on obtain some of its fundamental properties. introduce new definition the absolutely (2, 2)-summing operators from to , denote set such by $${\tilde{\Pi }}_2({\mathscr {E}},{\mathscr {F}})$$ with convention {E}})...