A functional model of a Lie algebra

Abstract

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