For an entirely non-selfadjoint operator with spectrum at zero, the imaginary component of which has an absolutely continuous spectrum (not necessarily dissipative and having lacunas in the spectrum), triangular and functional models are constructed.
This paper is devoted to a continuation of the author's research that was published in [1]; the subject is the study of the elementary Lie algebra of linear operators {A1, A2} in a Hilbert space H with the property that [A~., All = iA1.It is convenient to study the algebra {A1, A2} by using the Lie group of affine transformations of the line [3], whose Lie algebra of vector fields satisfies the same commutator relation. We limit ourselves to operators At and A2 for which the following assumptions are satisfied: a) A1 is a dissipative densely defined operator with the same defect spaces E = E+, (dim E = r < oo); b) A2 is bounded and (A2)IH C_ E, (At = (A -A*)/2i).The fundamental result of the the paper is that the Lie algebra {At, A2} can be realized in some space of meromorphic functions on a Riemann surface Q, and here one of the operators will be a multiplication by a meromorphic function f(P) -+ A(P)f(P), P 9 Q, while the second will be a translation operator, f(P) -+ f(a(P)), where a is an automorphism of the Riemann surface Q, (a 2 = 1, P 9 Q).1. A functional model of an operator A is usually understood as realization of a Hilbert space H in which the operator A acts a multiplication by an independent variable. By a functional model of the Lie algebra {A1, A~} we mean a realization of the space H in which at least one of the operators of the system {A1, A~} acts as a multiplication by an independent variable.
g+ S.(z) crn /' crz~S(z)g_ +g+ 9 H~_(E,o'ndl3) ~"We use L~(E, andl3) to denote the Hardy space of E-valued functions that are square integrable with respect to the measure ~rndt3 and correspond to the pole l-I = {z = c~ + i/3, -1 < a < 0}, while g~(E, andt3) is also a Hardy space, this time of functions of L~(E, andl3) that can be holomorphically continued in the half-plane + Re z > 0. The