If $A$ is a bounded linear operator and positive on a Hilbert space. Show that $A+\lambda I$ is an isomorphism $\forall\lambda>0$.

Question

Let $ H $ be a Hilbert space and let $ A\in \mathcal{B}(H) $ be a positive linear operator, that is, $\langle Ax,x \rangle \geq 0$ for all $ x\in H$. Show that $A+\lambda I$ is an isomorphism, for all $\lambda>0$.

There is an exercise with almost the same conditions, only that in that exercise it is shown that $A$ is an isomorphism. Could you say that since $A$ is an isomorphism then $A+\lambda I$ is also an isomorphism?

Without using the previous argument, how could you prove that $A+\lambda I$ is an isomorphism?

Answer 1

Yes you can use the previous exercise to solve this one, let $A' = A+\lambda I$, then $$\langle A'x , x\rangle = \langle Ax, x\rangle + \lambda \left\|x\right\|^2 \ge \lambda \left\|x\right\|^2$$ and this proves that $A'$ is an isomorphism using the previous exercise