Cordial greetings.
My first concern in this post is the following: we know that for real numbers $S,T$ is true $$\left(S\geq T\quad\mbox{and}\quad T\geq S\right)\Longrightarrow S=T.$$
Now, if $S,T$ are bounded linear operators defined in a Hilbert space $H$, is this still true?
While reading about operator theory in a book, they stated that if T is positive, then this operator is self-adjoint. I suppose this is trivial, but I have not been able to find the justification.