In the case that $L:B_1 \rightarrow B_2 $ is a linear mapping of Banach spaces and $L$ is a isometric isomorphism (bijection and $\|Lx\|_{B_1} = \|x\|_{B_2} $) can I say that $L\overline{L}= 1 $ is trivial ? (the bar denotes the complex conjugate);
TIA
Best Answer
I am going out on a limb to answer a different question – but possibly the question that was intended, if the comments are anything to go by:
Assume $L\colon H_1\to H_2$ is a linear isometry between Hilbert spaces. Then using the polarization identity $$ \langle x,y\rangle=\frac14\sum_{k=0}^3 i^k\lVert x+i^ky\rVert^2 $$ we can deduce $\langle Lx,Ly\rangle=\langle x,y\rangle$ for all $x$ and $y$, so that $L^*L=I_1$ (where $I_1$ is the identity on $H_1$). Since $L$ is also assumed to be a bijection, $LL^*=I_2$ follows, where $I_2$ is the identity on $H_2$.