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Systems of subspaces in Hilbert space that obey certain conditions, on their pairwise angles. (English. Russian original) Zbl 1285.46020

St. Petersbg. Math. J. 24, No. 5, 823-846 (2013); translation from Algebra Anal. 24, No. 5, 181-214 (2012).
For a Hilbert space \(H\) and a family of its subspaces \(H_i\,\,(i=1,\dots,n)\), the study of the system of subspaces \(S=(H,H_1,\dots,H_n)\), whose associated orthogonal projections satisfy certain relations is important in mathematical physics. The authors describe subspace systems by a construction of a system of subspaces in a Hilbert space on the basis of its Gram operator. They indeed study systems of subspaces \(H_1,\dots ,H_n\) of a complex Hilbert space \(H\) that satisfy the following conditions: for every index \(i>1\), the angle \(\theta _{1,i}\in (0,\pi /2)\) between \(H_1\) and \(H_i\) is fixed, the projections onto \(H_{2k}\) and \(H_{2k+1}\) commute for \(1\leq k\leq m\) (\(m\) is a fixed nonnegative number satisfying \(m\leq (n-1)/2\)) and all other pairs \(H_i\), \(H_j\) are orthogonal; see the survey by Yu. S. Samojlenko and A. V. Strelets [Ukr. Mat. Zh. 61, No. 12, 1668–1703 (2009); translation in Ukr. Math. J. 61, No. 12, 1956–1994 (2009; Zbl 1224.46044)].

MSC:

46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
47A46 Chains (nests) of projections or of invariant subspaces, integrals along chains, etc.
16G60 Representation type (finite, tame, wild, etc.) of associative algebras

Citations:

Zbl 1224.46044
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References:

[1] I. M. Gel\(^{\prime}\)fand and V. A. Ponomarev, Problems of linear algebra and classification of quadruples of subspaces in a finite-dimensional vector space, Hilbert space operators and operator algebras (Proc. Internat. Conf., Tihany, 1970) North-Holland, Amsterdam, 1972, pp. 163 – 237. Colloq. Math. Soc. János Bolyai, 5.
[2] J. J. Graham, Modular representations of Hecke algebras and related algebras, Ph. D. thesis, Univ. Sydney, 1995.
[3] P. R. Halmos, Two subspaces, Trans. Amer. Math. Soc. 144 (1969), 381 – 389. · Zbl 0187.05503
[4] Stanislav Krugliak and Yuri\? Samo\?lenko, On complexity of description of representations of \ast -algebras generated by idempotents, Proc. Amer. Math. Soc. 128 (2000), no. 6, 1655 – 1664. · Zbl 0951.46027
[5] V. Ostrovskyi and Yu. Samoilenko, Introduction to the theory of representations of finitely presented *-algebras. I, Reviews in Mathematics and Mathematical Physics, vol. 11, Harwood Academic Publishers, Amsterdam, 1999. Representations by bounded operators. · Zbl 0947.46037
[6] V. S. Sunder, \? subspaces, Canad. J. Math. 40 (1988), no. 1, 38 – 54. · Zbl 0656.46017 · doi:10.4153/CJM-1988-002-0
[7] H. N. V. Temperley and E. H. Lieb, Relations between the ”percolation” and ”colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ”percolation” problem, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1549, 251 – 280. · Zbl 0211.56703 · doi:10.1098/rspa.1971.0067
[8] S. A. Krugljak and Ju. S. Samoĭlenko, Unitary equivalence of sets of selfadjoint operators, Funktsional. Anal. i Prilozhen. 14 (1980), no. 1, 60 – 62 (Russian).
[9] D. K. Fadeev, A homological interpretation of the resultant of two polynomials, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 28 (1972), 164 (Russian). Investigations on the theory of representations.
[10] Yu. S. Samoilenko and A. V. Strelets, On simple \?-tuples of subspaces of a Hilbert space, Ukraïn. Mat. Zh. 61 (2009), no. 12, 1668 – 1703 (Russian, with Russian summary); English transl., Ukrainian Math. J. 61 (2009), no. 12, 1956 – 1994. · Zbl 1224.46044 · doi:10.1007/s11253-010-0325-7
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