Return to Revisions

Firstly, with regard to the above comment, I imagine that the claim is that the spaces are isometric in a natural way and this is indeed the case. For the sake of simplicity, I will assume that $M$ is injective and is multiplication by the positive sequence $(\lambda_n)$ on $\ell^2$. Then a simple calculation shows that both spaces consist of the weighted $\ell^2$-spaces of sequences $(\xi_n)$ for which $\sum \lambda_n \xi_n^2 <\infty$. One can use the spectral theorem to reduce to this case, at least for operators with discrete spectrum. The general case is much the same except that,rather than sequences, one has to consider multiplication operators on $L^2$-spaces.