How do we call a multi-valued operator $L$ with $∃c>0:∀y\in Y:∃x\in X:(x,y)\in L\text{ and }\left\|x\right\|_X\le c\left\|y\right\|_Y$?

Question

Let $X,Y$ be $\mathbb R$-Banach spaces and $L$ be a multi-valued linear operator from $X$ to $Y$ with $$\exists c>0:\forall y\in Y:\exists x\in X:(x,y)\in L\text{ and }\left\|x\right\|_X\le c\left\|y\right\|_Y\tag1.$$

Question 1: I'm not familiar with the theory of multi-valued operators. Is there an established notion for a multi-valued operator $L$ satisfying $(1)$?

It seems to be some kind of "boundedness" of the inverse $L^{-1}$.

Answer 1

The symbol-string you provide says that $L$ is surjective and bounded, so calling it 'invertible' or 'open' are both possibilities (the hypotheses for the Open Mapping Theorem for linear operators are exactly that). However, there's an issue around set-valued mapping in that determining the pull-back and how that should be handled is a little trickier.

https://pdfs.semanticscholar.org/28d9/26269f12d8f5a4d1267e87aa11cc3a290255.pdf discusses an inverse-mapping theorem for set-valued operators in Banach spaces and might be useful to you.