When $A_y$ is invertible?

Question

Given $y\in C[0,1]$

Let $A_y:C[0,1]\rightarrow C[0,1]: x\mapsto xy$

When $A_y$ is invertible?

Could you please help.

Answer 1

It's invertible whenever $y$ is invertible as an element of the ring $C[0,1]$. In this case, it means that $y$ never vanishes on $[0,1]$.

Basically, you've got a commutative ring $R = C[0,1]$, an element $y \in R$, and your $A_y$ is just the ring endomorphism "multiplication by $y$". Then obviously multiplication by $y$ can be inverted if you can divide by $y$, ie, $1/y\in R$. In your case, assuming $A_y$ sending $x\mapsto xy$ really means pointwise multiplication of functions, then $1/y$ exists whenever $y$ never vanishes on $[0,1]$, so that you can really define $(1/y)(t) = 1/y(t)$.