Does $S\leq T$ imply $\|S\|_{op}\leq \|T\|_{op}$? Let $T:V\to V$ and $S:V\to V$ be bounded self adjoint operators on a complex Hilbert space $V$. Is it true that if $0\leq T-S$ then $\|S\|_{op}\leq \|T\|_{op}$ with respect to the operator norm? Thanks!

It is true when $0\leq T\leq S$.
By functional calculus on $C^*(1,S)$, we have 
$$S\leq \lVert S\rVert 1,$$
so $$0\leq T\leq \lVert S\rVert 1.$$
Again by functional calculus on $C^*(1,T)$,
$$\lVert T\rVert \leq \lVert S\rVert.$$
