The homework question that I am trying to solve is the following, there are many parts but below is a summary of the ones I am struggling with:
Let $X, Y$ be two Hilbert spaces. Show that if a linear bounded operator $T : X → Y$ is surjective then the composite operator $T T^* : Y → Y$ is invertible, in which $T^*$ denotes the Hilbert-adjoint operator to $T$
The main issue I am having is finding a way to begin this proof. I am not sure how to relate the fact that the linear operator being surjective means that the composite one is invertible. Are there any known theorems or suggested ways of approaching this proof? Thank you
Suppose $TT^*y = 0$. Then $$ 0 = \langle y,TT^*y \rangle_Y = \langle T^*y,T^*y \rangle_X = \|T^*y\|_X^2$$ so that $T^*y = 0$ too. Since $T$ is onto there exists $x \in X$ with $y = Tx$. In a similar manner deduce $$\|y\|_Y^2 = \langle y,y\rangle_Y = \langle y,Tx \rangle_Y = \langle T^*y,x \rangle_X = 0$$ so that $y = 0$. Thus $TT^*$ is one-to-one.