Suppose that $H$ is a Hilbert space and we have sequence (or net) of non-increasing orthogonal projections $$E_1\geqslant E_2\geqslant E_3\geqslant\dots$$ Then we have also an operator $\bigwedge E_i$ which is a projection on space $\bigcap E_i(H)$.
Problem. Show that $E_i\rightarrow \bigwedge E_i$ strongly without using orthogonal projections $\bot$.
In Kadison/Ringrose book (2.5.6) and (2.5.7) are devoted to this problem. (2.5.6) shows that for non-decreasing sequence (net) $E_i\rightarrow \bigvee E_i$ strongly and then (2.5.7) uses $\bot$ to show our case.
Why do I want avoid $\bot$?
- In general, proving things for $\bigwedge E_i$ is easier then for $\bigvee E_i$. Thus, this case is somehow different. I wonder why it is so.
Moreover, in a Banach space we can also compare projections (abandoning orthogonality) and talk about $\bigwedge E_i$. So how about strong convergence in such spaces. Does this result still holds?(Edit: This is invalid, because $\bigwedge E_i$ does not make sence)
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