THE GAP OF THE GRAPH OF A LINEAR TRANSFORMATION

Abstract

Assume that $A$ is a linear transformation from $\mathbb{C}^n$ into $\mathbb{C}^m$. The Gap of the graph of $A$ is

\[\theta(A_g)=\frac{||A||}{\sqrt{1+||A||^2}}.\]

Here $||A||$ is the operator norm of $A$. This is an extention of the result in [2], in which the first author used another  method to prove for the case of $m= n$.