J. OPERATOR THEORY c Copyright by Theta, 2002 47(2002), 219–243 AN INTEGRAL REPRESENTATION FOR SEMIGROUPS OF UNBOUNDED NORMAL OPERATORS P. RESSEL and W.J. RICKER Communicated by Florian-Horia Vasilescu Abstract. An integral representation for semigroups {Us}s∈S of unbounded normal operators in a Hilbert space H is presented which admits a signifi- cantly larger class of semigroups S than usual. In particular, S need not have a topology and so the traditional assumption that the functions s 7→ hUsx, yi, for suitable elements x, y ∈ H, are continuous is no longer a requirement. The classical spectral theorem for a single (unbounded) normal or selfadjoint oper- ator is a consequence of the main result; the point is that the techniques used do not rely on the fact that a normal operator has a spectral decomposition via its resolution of the identity. Keywords: Semigroup representation, positive definiteness, normal opera- tor, spectral theorem. MSC (2000): 47B15, 47D03. 1. INTRODUCTION The integral representation (via spectral measures) of 1-parameter semigroups of bounded selfadjoint or normal operators on a Hilbert space is well known; see [19] or [10], Chapter XXII. The extension to semigroups of unbounded selfadjoint or normal operators is also well understood, although the technical assumptions are more involved; see [5], [6], [9], [11], [14], [15], for example, and the refer- ences there in. Such results sometimes extend to semigroups S more general than [0, ∞), although there are usually some sort of topological constraints on S. These topological requirements of S are often transferred to the semigroup of operators {Us}s∈S by requiring that the C-valued function (1.1) s 7→ hUsx, yi , s ∈ S, is continuous, for suitable vectors x and y in the underlying Hilbert space H. The aim of this note is to exhibit an integral representation for semigroups {Us}s∈S of unbounded normal operators which admits a significantly larger class 220 P. Ressel and W.J. Ricker of semigroups S than is usually the case. In particular, S is not assumed to have any topology and so no continuity assumptions are required of the functions (1.1). The techniques used come from the theory of positive definite functions. It is time to be more precise. Let S be a commutative semigroup with a unit element e and an involution s 7→ s− (i.e. (s−)− = s and (st)− = s−t− for all s, t ∈ S). A function ρ : S → C which satisfies ρ(e) = 1 and ρ(st−) = ρ(s)ρ(t) for all s, t ∈ S is called a character of S. The set of all characters is denoted by S∗; it becomes a completely regular topological space when equipped with the pointwise convergence topology inherited from CS. Given a Hilbert space H, let L(H) be the space of all continuous linear operators of H into itself and N (H) denote the collection of all (not necessarily bounded) normal operators in H. If T ∈ N (H), then its domain is denoted by D(T ); it is always dense. T ∗ denotes the adjoint operator of T . Each element of N (H) is a closed operator. As general references we use [3] and [17]. If T1 and T2 are linear operators in H, then T1 ⊆ T2 means D(T1) ⊆ D(T2) and T1x = T2x for x ∈ D(T1). The closure of a closable operator T (see [3]) is denoted by T . A function α : S → [0, ∞) is an absolute value if α is symmetric (i.e. α(s−) = α(s) for s ∈ S), α(e) = 1 and α(st) 6 α(s)α(t) for s, t ∈ S. The family of all absolute values on S is denoted by A(S). Given a map U : S → N (H) and α ∈ A(S) define n \ o (1.2) Dα := x ∈ D(Us): kUsxk 6 α(s)kxk for all s ∈ S , s∈S S where Us denotes the value of U at s ∈ S. Then define Dc := Dα. α∈A(S) Definition 1.1. Let S be a commutative, unital semigroup with an invo- lution. A map U : S → N (H) is called a ∗-representation if: (i) Ue = I, the identity operator on H. ∗ (ii) Us− = Us , s ∈ S. (iii) UtUs ⊆ Ust with D(UtUs) = D(Ust) ∩ D(Us), s, t ∈ S. (iv) UtUs = Ust, s, t ∈ S. S (v) Dc = Dα is dense in H. α∈A(S) We will see in Section 2 that (i)–(v) are natural if {Us}s∈S is to have an T integral representation. From (1.2) and (v) of Definition 1 we see D(Us) is s∈S necessarily a dense subspace of H (as it contains Dc). Our main result is the following one. Theorem 1.2. Let S be a commutative, unital semigroup with involution and U : S → N (H) be a ∗-representation. Then there exists a unique selfadjoint Radon spectral measure E : B(S∗) → L(H), with supp(Ex) ⊆ S∗ compact if and only if x ∈ Dc, and Z (1.3) Usx = sb(ρ) dE(ρ)x, x ∈ D(Us), s ∈ S, S∗ An integral representation for semigroups 221 ∗ where sb : S → C is defined by sb(ρ) := ρ(s). Some explanation is in order. By B(S∗) is denoted the Borel σ-algebra of S∗, i.e. the smallest σ-algebra containing all the open subsets of S∗. To say the function E : B(S∗) → L(H) is a selfadjoint spectral measure means that E(S∗) = I, that each operator E(A), for A ∈ B(S∗), is selfadjoint, that E is multiplicative (i.e. E(A ∩ B) = E(A)E(B) for A, B ∈ B(S∗)), and that E is countably additive for the weak (equivalently, the strong) operator topology in ∗ L(H), that is, Ex,y : A 7→ hE(A)x, yi, for A ∈ B(S ), is a σ-additive, C-valued measure for x, y ∈ H. Since the values E(A), for A ∈ B(S∗), are orthogonal 2 projections we see Ex,x(A) = kE(A)xk is actually non-negative, for x ∈ H. To say E is Radon means Ex,x is a Radon measure (i.e. inner regular) for each x ∈ H; see [2], Chapter 2, for example. The selfadjointness of E implies 4Ex,y = Ex+y, x+y − Ex−y, x−y + iEx+iy, x+iy − iEx−iy, x−iy, x, y ∈ H, and so each complex measure Ex,y is also Radon (i.e. its variation is a Radon measure). For x ∈ H, the H-valued, σ-additive measure A 7→ E(A)x, for A ∈ B(S∗), is denoted by Ex. Its support, denoted by supp(Ex), is defined to be the support of Ex,x (see [2], p. 22 for the definition of support of a non-negative Radon measure). Finally, the integral in (1.3) exists in the usual sense of integration with respect to the vector measure Ex ([12]); this is made precise in the next section. The proof of Theorem 1.2 relies on the methods of positive definite functions, together with an earlier version of Theorem 1.2 known for ∗-representations of bounded normal operators ([16]). The reduction to the bounded case is possible because the closed subspaces Dα (that they have this property is not obvious from (1.2)) turn out to be invariant for each Us, s ∈ S, and the restriction of Us to Dα is a bounded normal operator. Moreover Dc, which also turns out to be a subspace of H, plays a crucial role because it gives a description, via intrinsic T properties of U, of the space of vectors x ∈ D(Us) for which Ex has compact s∈S support. The spectral measure E is first constructed on Dc and then extended to H by a continuity argument. Because the arguments are rather technical the proof is delayed until Section 3. In the following Section 2 we prefer to discuss Theorem 1.2 itself in combination with Definition 1.1 and some relevant examples. In particular, we indicate how Theorem 1.2 deviates significantly from other results in the literature which are of a “similar nature” yet, as will be seen, are actually quite different. In Section 4 we indicate how the classical spectral theorem for a single (unbounded) selfadjoint or normal operator is a consequence of Theorem 1.2; the point is that the proof of Theorem 1.2 nowhere uses the fact that a normal operator has a spectral decomposition via its resolution of the identity. This is an important feature in relation to “similar” results in the literature alluded to above, which actually use the spectral theorem in their proofs. k An important particular case is S = N0 (with the usual addition and the identity as involution). Theorem 1.2 then provides the answer to an operator- valued moment problem. The existence of positive definite non-moment functions 2 2 on N0 ([2], Theorem 6.3.4) leads to ∗-homomorphisms U : N0 → L(D), where L(D) is the algebra of all endomorphisms on some appropriate pre-Hilbert space D, for which there is no integral formula analogous to (1.3). For the class of perfect semigroups S this phenomenon cannot occur; see the final section. 222 P. Ressel and W.J. Ricker 2. THEOREM 1.2: DISCUSSION Consider a selfadjoint spectral measure E : B(S∗) → L(H). Every bounded Borel function f : S∗ → C is integrable with respect to the L(H)-valued measure E in the sense of vector measures ([7], [12]). This integral, denoted by R f dE, is an S∗ element of L(H).