Suppose $\mathcal{H}$ is a reproducing kernel Hilbert space. If the kernel is normalized then the multiplier algebra $\mathcal{M}$ is an algebra that is sitting inside $\mathcal{H}$. Is there any other algebra sitting inside RKHS with some conditions? If yes, what are all the algebras?
RKHS Definition Let $X$ be a set, we call $\mathcal{H} \subset \mathcal{F}(X,\mathbb{F})$ a RKHS on $X$ if
1.$\mathcal{H}$ is a vector subspace of $\mathcal{F}(X,\mathbb{F})$.
2.$\mathcal{H}$ is endowed with an inner product $\langle , \rangle $ with respect to which $\mathcal{H}$ is a Hilbert space.
3.$\forall$ $x\in X$, the evaluational functional $E_x: \mathcal{H} \rightarrow \mathbb{F}$ defined by $E_x(f)=f(x)$ is bounded.
