"bounding" an unbounded operator

I was wondering if, given a certain unbounded operator on a Hilbert space, it can (naively speaking) be "cutted" (or "bounded") by certain projections.

So, thinking about this in a more sensible way, I have the following question:

Let $T$ be a unbounded (densely defined) self-adjoint (positive) operator on $\mathcal{H}$, and let $\{P_\lambda\}_{\lambda > 0}$ the spectral family of $T$. Then we can look at the following: $\mathcal{H}_\lambda:= P_{(\lambda^{-1},\lambda)}\mathcal{H}$, where $P_{(\lambda^{-1},\lambda)}$ correspond to the Borel functional calculus on $T$ of the characteristic function on the interval $(\lambda^{-1},\lambda)$.

Is it true that, on $\mathcal{H}_\lambda$, $||Tx||\leq \lambda||x||$ ?

Solution 1:

Yes, of course.

The functional calculus preserves positivity. The function $(\lambda-t)\,1_{(0,\lambda)}(t)$ is non-negative, so the operator $(\lambda\,I-T)\,P_{(0,\lambda)}$ is positive. That is, $T\,P_{(0,\lambda)}\leq\lambda\,P_{(0,\lambda)}$. In particular, $\|T\,P_{(0,\lambda)}\|\leq\lambda$.