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.


47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
46C99 Inner product spaces and their generalizations, Hilbert spaces