# Understanding topological divisor of zero

I am reading this paper also here where in the Theorem 2.1 term topological divisor of zero has been used. I have gone through wiki articles where it has been mentioned that

An element $z$ of a Banach algebra $A$ is called a topological divisor of zero if there exists a sequence $x_1$, $x_2$, $x_3$, ... of elements of $A$ such that the sequence $zx_n$ converges to the zero element, but the sequence $x_n$ does not converge to the zero element.

However, I am not able to apply this definition in the paper mentioned above. Is there any other definitions of topological divisor of zero? Please explain me about this term so that I can understand why author of the paper has used this term?

If your Banach algebra $A$ is the algebra $\mathcal L(E)$ of all bounded linear operators on some Banach space $E$, then one can use the following characterization: an operator $T\in\mathcal L(E)$ is $not$ a right topological divisor of $0$ if and only if $T$ is one-to-one with closed range; equivalently, if there is a constant $c>0$ such that $\Vert Tx\Vert\geq c\Vert x\Vert$ for all $x\in E$.
• I think there is a misleading statement in Theorem 2.1. What the author wanted to say (I hope) is that an operator $T$ is not a right topological divisor of $0$ if and only if the $adjoint\; operator$ $T'$ is onto. – Etienne Jun 13 '13 at 7:06