ABOUT POLYNOMIAL SEMILINEAR RELATIONS

Pairs of self-adjoint linear operators in a Hilbert space which satisfy semilinear relations arise in different problems of mathematics and physics (see, for exam-ple [1]) and were studied in [4, 5] and others. Following [4, 5] we will say, that a pair of bounded selfadjoint operators (A,B) in a Hilbert space H satisfy semilinear relation if m∑ i=1 f i(A)B g i(A) = h(A), where f i(t), g i(t), h(t) are polynomials on R for all i = 1,m. With any semilinear relation the following three objects are associated: • a characteristic function, which is a polynomial of two variables on R2: Pn(t, s):= m∑ i=1 f i(t)g i(s)