Just started to learn about linear operator theory, and trying to understand adjoint operator. Here's a conceptual problem, can someone help me to clarify? Thanks
Let L be a bounded linear operator on a Hilbert space H. Verify the following relationships: $ker(L)=ker(LL^*)$ and $$cl(im(LL^*))=cl(im(L))$$
where cl is the closure and ker, and im are kernel and image.
Clearly $\mathcal{N}(L^{\star})\subseteq\mathcal{N}(LL^{\star})$. Conversely, if $x\in\mathcal{N}(LL^{\star})$, then $x\in\mathcal{N}(L^{\star})$ because $$ 0 = (LL^{\star}x,x) = (L^{\star}x,L^{\star}x)=\|L^{\star}x\|^{2}. $$ Therefore $$ \mathcal{R}(L)^{\perp}=\mathcal{N}(L^{\star})=\mathcal{N}(LL^{\star})=\mathcal{R}((LL^{\star})^{\star})^{\perp}=\mathcal{R}(LL^{\star})^{\perp} $$ Denoting closure with a 'c' superscript, $$ \mathcal{R}(L)^{c}=\mathcal{R}(L)^{\perp\perp}=\mathcal{R}(LL^{\star})^{\perp\perp}=\mathcal{R}(LL^{\star})^{c}. $$