Concerning this: "I am likewise interseted in the operator algebra of operators on S* that restrict to operators on S. In particular, I would like to abstractly characterize this space"

I am not sure that I understand you correctly, but if you want to characterize the operators which are extensions from S to S*, then the following can be the answer. First, let us use some notations: for any x,y$\in$S we set $(x,y)=\int x(t)y(t)dt$, for any operator (everywhere operator will be linear and continuous) $A:S\to S$ we define a transposed operator $A^T:S\to S$ by formula $(Ax,y)=(x,A^Ty)$ (it does not always exist), the dual operator A*:S*$\to$ S* by formula Af(x)=f(Ax) (it always exists) and for any operator B:S$\to$S* its dual B*:S$\to$S by formula Bx(f)=Bf(x) (it also always exists). Let us also consider an operator F:S$\to$S, Fx(y)=(x,y).

Then

1. B:S*$\to$S* is an extension of $A:S\to S$ iff $B\circ F=F\circ A$;

2. an operator A:S$\to$S can be extended to some operator B*:S$\to$S iff there exists a transposed operator $A^T:S\to S$; in this case B=A$^T$*;

3. an operator B*:S$\to$S is an extension of some operator A:S$\to$S iff for B* there exists a transposed operator B*$^T:S\to S$; in this case A=B*$^T$.

Or were you asking about something else?

Concerning this: "I am likewise interseted in the operator algebra of operators on S* that restrict to operators on S. In particular, I would like to abstractly characterize this space"

I am not sure that I understand you correctly, but if you want to characterize the operators which are extensions from S to S*, then the following can be the answer. First, let us use some notations: for any x,y$\in$S we set $(x,y)=\int x(t)y(t)dt$, for any operator (everywhere operator will be linear and continuous) $A:S\to S$ we define a transposed operator $A^T:S\to S$ by formula $(Ax,y)=(x,A^Ty)$ (it does not always exist), the dual operator A*:S*$\to$ S* by formula Af(x)=f(Ax) (it always exists) and for any operator B:S$\to$S* its dual B*:S$\to$S by formula Bx(f)=Bf(x) (it also always exists). Let us also consider an operator F:S$\to$S, Fx(y)=(x,y).

Then

1. B:S*$\to$S* is an extension of $A:S\to S$ iff $B\circ F=F\circ A$;

2. an operator A:S$\to$S can be extended to some operator B:S*$\to$S* iff there exists a transposed operator $A^T:S\to S$; in this case B=A$^T$*;

3. an operator B:S*$\to$S* is an extension of some operator A:S$\to$S iff for B* there exists a transposed operator B*$^T:S\to S$; in this case A=B*$^T$.

Or were you asking about something else?