Is this operator surjective?

Question

Let $H$ Hilbert space, $A \colon D(A) \subset H \to H$ be a linear, closed, densely defined operator.

In [Fabbri, Giorgio, Fausto Gozzi, and Andrzej Swiech. "Stochastic optimal control in infinite dimension." Probability and Stochastic Modelling. Springer (2017).] at p.175 it is claimed that it is well known that $B=(I+AA^*)^{-1/2}$ is bounded, strictly positive, selfadjoint and $A^*B$ is a bounded linear operator.

Now $(I+AA^*)$ has empty kernel, so it is injective, how do you show it is surjective?

In this way we can take the inverse $(I+AA^*)^{-1}$ and square it in order to obtain $B$.

Answer 1

Hint: Let $T=I+AA^*$.

1. Show that $\langle Tx, x\rangle \ge \|x\|^2$ for all $x\in D(T)$.

2. Show that the range $R(T)$ is closed using 1.

3. Show that $R(T)$ is dense.