I am reading a book Gustafson-Rao's Numerical Range and come across a problem that I really don't understand.
In theorem 2.1-2 of the book, it asserts that for an operator valued function $$F(z)=(I-zT)^{-1}$$ where $|z|<1$ and $T$ is a linear operator on a Hilbert space $H$ with spectral radius less than 1, there exists a Hilbert space $K,s.t \;H\subset K$ and a unitary $U$ in $K$ s.t. $$F(z)=P(I_K+zU)(I_K-zU)^{-1},|z|<1,$$ since $F(z)$ is holomorphic in the disk $|z|<1$,$F(0)=I$ and Re$F(z)\geq O$ (I even do not know what this inequality means..) by a theorem of Riesz, generalized to operator valued function. $P$ is a projection from $K$ to $H$, and $I_K$ is the identity map on $K$.
So far, I only know the Riesz representation theorem for a Hilbert space and its dual. Is the generalized Riesz theorem starting from this one? I have searched a while and actually check the original proof of the theorem 2.1-2 in a paper but it seems the author of the book simply copied the solution without much change so no any clue of what the generalzed Riesz theorem is.
If there is indeed one, what kind of knowledge do I need to understand it? what does the theorem say? I only know some basic functional analysis.