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On sequences of projections. (English) Zbl 0622.47019
The following statement is shown:
Let $$t\in [0,1]$$ and let $$e_ 1,e_ 2,..$$. be nonzero projections on a Hilbert space satisfying the relations
(a) $$e_ ie_{i\pm 1}e_ i=te_ i,$$
(b) $$e_ ie_ j=e_ je_ i$$ for $$| i-j| \geq 2.$$
If 4 cos$${}^ 2(\pi /(m+1))<1/t<4 \cos^ 2(\pi /(m+2))$$, then there exist at most 2m-1 such projections. If (b) is replaced by $$e_ ie_ j=0$$, the upper bound is m. It is known that there exists an infinite sequence of projections with (a) and (b) for all other values of t.

##### MSC:
 47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.) 46C99 Inner product spaces and their generalizations, Hilbert spaces
##### Keywords:
operator algebras; projections; positivity