We consider a collection of self-adjoint commuting operators A=An, nεN∼(N), which are connected with a non-self-adjoint operator B by relations of the form AnB=BFn(A),nεN, under the condition imposed on B of the formB * B-F0 (BB*, A). We prove that the problem of description of representations of such relations can be reduced to the case of a unitary B.
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Yu. M. Berezanskii, V. L. Ostrovskii, and Yu. S. Samoilenko, “Eigenfunction expansion of families of commuting operators and representations of commutation relations,” Ukr. Mat. Zh.,40, No. 1, 108–109 (1988).
V. L. Ostrovskii and Yu. S. Samoilenko, “An application of the protective spectral theorem to non-commutative families of operators,” ibid., No. 4, 421–433 (1988).
Yu. M. Berezanskii and Yu. G. Kondrat'ev, Spectral Methods in Infinite Dimensional Analysis [in Russian], Naukova Dumka, Kiev (1988).
V. L. Ostrovskii and Yu. S. Samoilenko, “Unbounded operators satisfying non-Lie commutation relations,” Rep. Math. Phys.,28, No. 3, 93–106 (1989).
V. L. Ostrovskii and Yu. S. Samoilenko, “Families of unbounded self-adjoint operators connected by non-Lie relations,” Funkts. Anal. Prilozhen.,23, No. 2, 67–68 (1989).
É. E. Vaisleb and Yu. S. Samoilenko, “On a representation of the relations AU=UF(A) by an unbounded and a unitary operator,” in: Boundary Problems for Differential Equations [in Russian], Institute of Mathematics, Academy of Sciences of the Ukrainian SSR, Kiev (1988), pp. 30–52.
É. E. Vaisleb and Yu. S. Samoilenko, “Representations of operator relations by unbounded operators, and many dimensional dynamical systems,” Ukr. Mat. Zh.,42, No. 8, 1011–1019 (1990).
V. L. Ostrovskii, “Representations of a family of quadratic algebras with three generators,” in: Applications of Mathematical Analysis in Mathematical Physics [in Russian], Institute of Mathematics, Academy of Sciences of the Ukrainian SSR, Kiev (1989, pp. 94–103.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 9, pp. 1258–1262, September, 1990.
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Vaisleb, É.E. Representations of relations connecting a collection of commuting self-adjoint operators with a non-self-adjoint one. Ukr Math J 42, 1119–1123 (1990). https://doi.org/10.1007/BF01056608