I am not well versed in functional analysis or operator theory, but I've been pondering about this operator equation for quite some time and cannot put my head around it. So maybe you could nudge me in the right direction?
Given the arbitrary operator $T:\mathcal{F}\rightarrow\mathbb{R}$ over some function space $\mathcal{F}$, I want to find all functions $g\in\mathcal{F}$ s.t. $Tg=Tf+c$ for any function $f\in\mathcal{F}$ and constants $c\in\mathbb{R}$.
From the equation it would seem immediate to get $T^{-1}$ and call it a day. There are conditions for $T$ to be invertible, I know, but are there explicit methods to find $T^{-1}$ given $T$? Does $T$ need to be more specific to have some form of explicit formulae?
Another approach I had was to set $g=Uf$ with $U:\mathcal{F}\rightarrow\mathcal{F}$. Then $TUf-Tf=c$ which resembles some form of derivative. In that train of thought I wondered if there is some form of variational Taylor approximation $TUf=Tf+\delta_{Uf} T$ which would lead to $\delta_{Uf}T=c$? This reminded me of the basic idea behind natural gradient methods wherein you try to find the minimal gradient, say $\delta_{Uf}T$, constrained not be smaller/higher than $c$.
So basically any hint would be appreciated. Maybe it falls short due to my lack of knowledge in the field - if so please say.
First, $T$ is not invertible unless $\mathcal F=\mathbb R$, so calculating $T^{-1}$ is not a right approach.
Hint: For calculating all solutions $g$ you need to find the following.