# Journal of Operator Theory

Volume 73, Issue 2, Spring 2015 pp. 425-432.

Quotients of adjointable operators on Hilbert $C^{*}$-modules

**Authors**:
Marzieh Forough

**Author institution:**School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

**Summary: **Let $T$ and $S$ be bounded adjointable operators on a Hilbert
$C^*$-module $E$ such that $\mathrm{ker}(S)$ is orthogonally complemented in
$E$ . We prove that the quotient $TS^{-1}$ is a closed operator with
orthogonally complemented graph in $E \oplus E$ if and only if
$\mathrm{ran}(T^{*})+\mathrm{ran}(S^{*})$ is closed. We mean here by
$S^{-1}$ the inverse of the restriction of $S$ to
$\mathrm{ker}(S)^{\perp}$. This leads us to study the operators as
$TS^{\dag}$, whenever $S$ admits the Moore--Penrose inverse $S^{\dag}$. Note
that in case of an injective Moore--Penrose invertible operator $S$, we have $S^{-1}=S^{\dag}$. Then we present some applications of these results. Moreover, the quotients of regular operators are also investigated in this paper.

**DOI: **http://dx.doi.org/10.7900/jot.2014jan28.2010

**Keywords: **bounded adjointable operators, regular operators, Hilbert $C^{*}$-modules, quotient of operators, Moore--Penrose inverses

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