The spectral properties of certain linear operators and their extensions

Let $H$ be a Hilbert space with inner-product $(x,y)$, and let $R$ be a bounded positive operator on $H$ which determines an inner-product, $\langle x,y\rangle=(Rx,y), x, y\in H$. Denote by $H^-$ the completion of $H$ with respect to the norm $\|x\|=\langle x,x\rangle^{1/2}$. In this paper, operators having certain relationships with $R$ are studied. In particular, if $T=SR^{1/2}$ where $S\in B(H)$, then $T$ has an extension $T^-\in B(H^-)$, and $T$ and $T^-$ have essentially the same spectral and Fredholm properties.