All Questions

Let $\mathcal H$ be a Hilbert space and let $O$ be an operator. Obviously $M=O^\dagger O$ is a semi-positive definite operator and $v\in\ker M$ if and only if $v\in\ker O$. Therefore it seems to me ...

Let $H$ be a separable Hilbert space over $\mathbb{R}$. Let ${A} = \{a\colon H\to H\,|\,a\text{ is a positive, compact linear operator}\}$. By the spectral theorem, given $a \in A$, there are scalars $...

Let us suppose that $A,A_1,A_2,\ldots$ are non-negative definite self-adjoint bounded linear operators in $L(\mathbb H)$, where $\mathbb H$ is a separable Hilbert space. $(v_j)_{j\ge1}$ and $(\...

I need to show one of the two following equivalent results. If true, it must be a simple proof but I do not seem to be able to make it work. Thank you in advance. 1) Consider a continuous symmetric ...