The product of operators with closed range in Hilbert C*-modules

TL;DR

This paper characterizes when the product of two bounded adjointable operators with closed ranges in Hilbert C*-modules also has a closed range, using conditions involving kernels, ranges, and angles between submodules.

Contribution

It provides necessary and sufficient conditions for the closedness of the product operator’s range in terms of orthogonal summands and angles in Hilbert C*-modules.

Findings

$TS$ has closed range iff $Ker(T)+Ran(S)$ is an orthogonal summand.

$TS$ has closed range iff $Ker(S^*)+Ran(T^*)$ is an orthogonal summand.

Positive Dixmier angle condition ensures $TS$ has closed range.

Abstract

Suppose $T$ and $S$ are bounded adjointable operators with close range between Hilbert C*-modules, then $TS$ has closed range if and only if $Ker(T)+Ran(S)$ is an orthogonal summand, if and only if $Ker(S^*)+Ran(T^*)$ is an orthogonal summand. Moreover, if the Dixmier (or minimal) angle between $Ran(S)$ and $Ker(T) \cap [Ker(T) \cap Ran(S)]^{\perp}$ is positive and $ \bar{Ker(S^*)+Ran(T^*)} $ is an orthogonal summand then $TS$ has closed range.