# “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.

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 $$\mathcal{H}_\lambda$$, $$||Tx||\leq \lambda||x||$$ ?
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, $$\|T\,P_{(0,\lambda)}\|\leq\lambda$$.