What do we mean by saying product of operators is a scalar?

The motivation for this question is considering the helicity of a massless particle in relativistic QFT.

As the definition for helicity operator $h$
$$
h=frac{mathbf{J} cdot mathbf{P}}{|mathrm{P}|}
$$

where $J$ and $P$ are three dimensional angular momentum operators and momentum operators respectively.

Now every textbook says that this is a "scalar" under Lorentz transformations for massless particles, which follows from the definition of rotations and boosts.

Now

1)Firstly I just want to be more precise to ask that in what sense do we mean by "scalar" for calling this quantity, since it is a operator rather than a number. Does it mean that the $h$ operator does not change its form under Lorentz transformation?

2)Secondly it is easy to prove that $h$ is invariant under rotations, but how do we prove that it is invariant under boosts? Or do they?
In fact in another question there is an answer saying that the boosts generators do not commute with h in fact

$$
left[frac{J cdot P}{H}, K_{i}right]=ileft(frac{epsilon_{i j k} K_{j} P_{k}}{H}+J_{i}-frac{P_{i}}{H^{2}} J cdot Pright)
$$

But on the other hand, massless particles always travel at the speed of light, so my guess is that boosts do not have influence on the massless particles but only rotations do, is this argument the reason why helicity is an invariant quantity for massless particles? And is there any more solid or inspiring explanation on this thing.