Ostrovskij, V. L.; Samojlenko, Yu. S. Representations of \(\ast\)-algebras with two generators and polynomial relations. (English. Russian original) Zbl 0779.47033 J. Sov. Math. 59, No. 5, 1107-1113 (1992); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 172, 121-129 (1989). See the review in Zbl 0709.47041. MSC: 47C10 Linear operators in \({}^*\)-algebras 16G60 Representation type (finite, tame, wild, etc.) of associative algebras 46K10 Representations of topological algebras with involution Keywords:generators of \(\ast\)-algebras; structure of pairs of self-adjoint operators in Hilbert space; tame representations; wild representations; all \(\ast\)-algebras with quadratic relations are tame; classification; representations; algebras with general polynomial relations Citations:Zbl 0673.00016; Zbl 0709.47041 PDFBibTeX XMLCite \textit{V. L. Ostrovskij} and \textit{Yu. S. Samojlenko}, J. Sov. Math. 59, No. 5, 1107--1113 (1989; Zbl 0779.47033); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 172, 121--129 (1989) Full Text: DOI References: [1] E. K. Sklyanin, ?Algebraic structures connected with the Yang-Baxter equation. II. Representations of the quantum algebra,? Funkts. Anal. Prilozhen.,17, No. 4, 34?48 (1983). [2] A. M. Vershik, ?Algebras with quadratic relations,? in: Spectral Theory of Operators and Infinite-Dimensional Analysis [in Russian], Inst. Mat., Akad. Nauk UkrSSR, Kiev (1984), pp. 32?57. [3] E. Nelson, ?Analytic vectors,? Ann. Math.,70, No. 2, 572?615 (1959). · Zbl 0091.10704 · doi:10.2307/1970331 [4] M. Flato, J. Simon, H. Snellman, and D. Sternheimer, ?Simple facts about analytic vectors and integrability,? Ann. Sci. Ec. Norm. Sup., 4 ser.,5, No. 3, 424?434 (1972). · Zbl 0239.22019 [5] Yu. S. Samoilenko, Spectral Theory of Collections of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1984). [6] Yu. M. Berezanskii, V. L. Ostrovskii, and Yu. S. Samoilenko, ?Decomposition in eigenfunctions of families of commuting operators and representations of commutation relations,? Ukr. Mat. Zhurn.,40, No. 1, 106?109 (1988). · doi:10.1007/BF01056461 [7] V. L. Ostrovskii and Yu. S. Samoilenko, ?Application of the projectional spectral theorem to noncommuting families of operators,? Ukr. Mat. Zhurn.,40, No. 4, 469?481 (1988). [8] P. Halmos, Hilbert Space in Problems [Russian translation], Moscow (1970). · Zbl 0204.15001 [9] É. E. Vaisleb and Yu. S. Samoilenko, ?Representation of relations AU=UF(A) by unbounded self-adjoint and unitary operators,? in: Boundary Problems for Differential Equations [in Russian], Inst. Mat., Akad. Nauk UkrSSR, Kiev (1988). [10] A. N. Sharkovskii, Yu, L. Maistrenko, and E. Yu. Romanenko, Difference Equations and Their Applications [in Russian], Naukova Dumka, Kiev (1986). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.