Quick trivial question that I did not find an answer to immediately from a textbook, so good material for stackexchange.
In operator theory what does it mean for an operator to be a subset of other operator e.g. $A \subset B$? We are considering dense operators, as bonus does this mean the range, domain or both are dense?
The meaning is actually quite simple. When we write $A\subset B,$ what we mean is that the graph of $A$ is a subset of the graph of $B$. In particular, if $A:D(A)\subset H\rightarrow H$ and $B:D(B)\subset H\rightarrow H,$ then we mean that $D(A)\subset D(B)$ and $Ax=Bx$ for all $x\in D(A).$