Criteria for the existence of systems of subspaces related to a certain class of unicyclic graphs
DOI:
https://doi.org/10.37863/umzh.v73i4.6354Keywords:
.Abstract
UDC 512.552.4
We study the configurations of subspaces of a Hilbert space associated with a unicyclic graph, which is a cycle of length m⩾ and has, at each vertex of the cycle, a chains of length s\geqslant 1 glued to the vertex.
There is a one-to-one correspondence between the vertices and subspaces.
If an edge connects two vertices, then the angle between subspaces is equal to \psi\in(0;\pi/2), otherwise the subspaces are orthogonal.
Applying the theorem on reduction of unicyclic graph, we prove that nonzero configurations exist if and only if \cos\psi\in(0;\tau_{m,s}].
We obtain formulas for \tau_{m,s} and show that~\bigcap\limits_{m,s}(0;\tau_{m,s}] = (0;2/5].
References
Yu. S. Samojlenko, A. V. Strelecz, O prosty`kh n-kakh podprostranstv gil`bertova prostranstva, Ukr. Math. J., 61, № 12, 1668 – 1703 (2009).
M. A. Vlasenko, N. D. Popova, O konfiguracziyakh podprostranstv gil`bertova prostranstva s fiksirovanny`mi uglami mezhdu nimi, Ukr. Math. J., 56, № 5, 606 – 615 (2004). DOI: https://doi.org/10.1007/s11253-005-0074-1
H. Wenzl, On sequences of projections, C. R. Math. Acad. Sci. Soc. R. Can., 9, № 1, 5 – 9 (1987).
N. D. Popova, On finite-dimensional representations of one algebra of Temperley – Lieb type, Methods Funct. Anal. and Topology, 7, № 3, 80 – 92 (2001).
N. D. Popova, O. V. Strilecz`, Pro sistemi pidprostoriv gil`bertovogo prostoru, shho pov'yazani z unicziklichnim grafom, Zb. pracz` In-tu matematiki NAN Ukrayini, 1, № 1, 166 – 177 (2015).
