On sequences of projections. (English) Zbl 0622.47019

Wenzl, Hans

C. R. Math. Acad. Sci., Soc. R. Can. 9, 5-9 (1987).

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