Commutative algebras of unbounded operators

Abstract

We consider algebras U of unbounded operators defined on a common, invariant, dense domain D in a Hilbert space H. We assume that U is generated by a finite number of operators S 1,…, S d satisfying the inequality, ∑ x ∑ β 〈S βX x|S xX β〉⩾0 for all finite sets of vectors x α indexed by α = ( α 1,…, α d ), α i = 0,1,…, 1⩽ i ⩽ d, where S α = S α 1 1…S α d d, and >·¦·> denotes the inner product in H. We show that every such algebra dilates to a commutative algebra in a, generally bigger, Hilbert space. The latter algebra is generated by commuting, formally normal, extension operators, N 1,…, N d such that, S j x = N j x, x ϵ D, 1 ⩽ j ⩽ d. Moreover, the adjoint operators N ∗ j are represented as multiplication operators on a quotient of the space of D-valued polynomials in complex variables ( z 1,…, z d )∈ C d (N j ∗)(z j,…, z d) = z ju(z j,…, z d) . If it is further assumed that D contains a dense space of vectors, analytic for each of the operators S j , then we prove that the N j 's are essentially normal, and there is a spectral measure E on the product of the spectrum of the closed normal operators N ̄ j such that N ̄ i = f λ, dE λ , and S j x = ∝ λ j dE λ x for x ϵ D. As an application, we consider representations of the canonical commutation relations in d degrees of freedom. If P 1,…, P d , Q 1,…, Q d , denote the momentum (resp. position operators), and S j = P j + iQ j , we show that S 1,…, S d , admits a representation as above. We also show, using our analytic vector theorem, that the corresponding extension operators N ̄ 1,…, N ̄ d , are commuting and normal for the Schrödinger representation. Finally, we calculate the spectral representation for the operators N ̄ j and prove absolute continuity, and uniform multiplicity equal infinity. The exposition of the paper begins with the case d = 1. This is the unbounded version of the subnormal operators which have been studied extensively in the past in the case of bounded operators. We stress the differences between the theory for the bounded, and the unbounded case. Even in the case d = 1, the applications of the unbounded theory are surprising.