Question about the notation $S \subset T$ ,where $S$ and $T$ are operators

Question

I want to prove that if $S\subset T$. Then $T^{*}\subset S^{*}$.

But what does $S\subset T$ mean? $S$ and $T$ are operators and not sets.. :/

Answer 1

It is standard notation and means that $T$ is an extension of $S$, i.e. $D(S)\subset D(T)$ and $T|_{D(S)}=S$ where $D(S)$ denotes the domain of $S$ (even shorter: $\Gamma(S)\subset\Gamma(T)$ where $\Gamma(S)$ is the graph of $S$).