Are there conditions for a sum or difference of linear maps between the same spaces to be an isomorphism?

Question

Suppose we have two linear maps $L:V\rightarrow W$ and $M:V\rightarrow W$, where $\mbox{dim}\hspace{0.05cm}V=\mbox{dim}\hspace{0.05cm} W$ and $L$ is an isomorphism. Are there any sufficient conditions we can place on $M$ to guarantee that $L-M$ is an isomorphism?

Answer 1

This is not complete solution, but just some interesting notes. I will work over $\mathbb{R}$ or $\mathbb{C}$. If we have $||L^{-1}M||<1$, then we can create an inverse, denoting $L^{-1}M=X$, by $(\sum_i X^i)L^{-1}$. Moreover, if $||M^{-1}L||<1$, the same aurguement applies. In particular, if we have $||L^{-1}||^{-1}<||M||$ or $||M^{-1}||^{-1}<||L||$. Sadly we can have $||M^{-1}||>||M||^{-1}$, so we cannot go much further with this route.

Note that it is clear than an inverse exists if and only if $char(L^{-1}M)(1)\neq 0$, so that failure is actually pretty rare, namely, there is only a finite number of values of $a$ such that $L-aM$ is not invertible.

Going a different direction, we note that $I-L$ is invertible if and only if $A(I-L)A^{-1}=I-ALA^{-1}$ is, so taking $L=diag(1, \gamma_1, \dots \gamma_n)$, $\gamma_1\neq 0$, we get a pretty large set of matrices without the condition.