Let $M=A-B$ be a symmetric matrix of order n.
I know $\lambda_{max}=\sup_{x\neq 0} \frac{x^tMx}{x^t x}$. Where $ x\in R^n$.
Can I write it like $\lambda_{max}=\sup_{x\neq 0} \frac{x^tAx}{x^t x}-\inf_{y\neq 0} \frac{y^tBy}{y^t y}$?
Note that my confusion is about vectors $x$ and $y$. Should they remain same or can they be different?
If we simply derive the computation you suggest we get: \begin{align*} \lambda_\max (M) &= \sup_{x\neq0} \frac{x^\top M x}{x^\top x}\\ &= \sup_{x\neq0} \frac{x^\top (A-B) x}{x^\top x}\\ &= \sup_{x\neq0} \left( \frac{x^\top A x}{x^\top x} - \frac{x^\top B x}{x^\top x} \right) \end{align*} But you cannot separate the $\sup$. Always try to think of simple examples in $\mathbb{R}$, such as
$$\sup_{x \in \mathbb{R}} f(x) + g(x) \leq \sup_{x \in \mathbb{R}} f(x) + \sup_{y \in \mathbb{R}} g(y)$$ this holds because you basically have "more freedom" on the right-hand side.