Is there any property of the operator on a Hilbert space that we loose when we study a unitarily equivalent operator and prove something about that one instaed?
I cant think of any, to me this looks like the same thing as a change of basis on finite dimension i.e just a new matrix. But maybe therse is a significant difference between changing spaces then just changing basis, and between finite and infinite dimension in this aspect.
I would say the situation is similar in Hilbert spaces of finite and infinite dimension. A unitary transformation preserves the inner product, so the space is unchanged as a Hilbert space. Infinite dimensional spaces might look different despite being "equal" (isometric). There are issues related to contiuty of linear maps in infinite dimensional spaces, but I think that is beside the point.
There typically is no problem if you study a single operator with no other objects of interest. But if some basis (or way of representing the space) is more suitable than others, then changing the basis can be a nuisance. And most importantly, you cannot always make several operators nice with the same unitary transformation — just like you cannot simultaneously diagonalize two hermitean matrices unless they commute.
In short, it depends on what you want to know about your operator. If you want to study it in isolation, then unitary transformations don't usually matter. If you don't, they might.