Krein space unitary dilations of Hilbert space holomorphic semigroups

The infinitesimal generator $A$ of a strongly continuous semigroup on a Hilbert space is assumed to satisfy that $B_\beta:=A-\beta$ is a sectorial operator of angle less than $\frac{\pi}{2}$ for some $\beta \geq 0$. If $B_\beta$ is dissipative in some equivalent scalar product then the Naimark-Arocena Representation Theorem is applied to obtain a Kre\u{\i}n space unitary dilation of the semigroup.


Introduction
A Kreȋn space unitary dilation of a strongly continuous semigroup {T (t)} on a Hilbert space H was built up by B. McEnnis under the assumption that the numerical range of the infinitesimal generator A lies on a sector of semi-angle 0 < θ < π 2 around the real line [10]. McEnnis' construction follows the one given by C. Davis for uniformly continuous semigroups [5]. Both constructions lean on the existence of a selfadjoint operator G such that d dt for all t > 0 and all h ∈ H. The key element in McEnnis' is the convexity of the numerical range (the Haussdorff-Toeplitz Theorem) which grants the mθ-dissipativeness of A − β for some β ≥ 0 and, in consequence, a contractive holomorphic extension of {e −βt T (t)} within the sector |arg(z)| < π 2 − θ. The sectoriality of the numerical range can be replaced by sectoriality of the spectrum together with some norm estimates of the resolvent. The latter conditions allow a functional calculus which, in turn, gives a one-to-one correspondence between the type of closed operators A satisfying those constraints and the bounded holomorphic strongly continuous semigroups {T (z)} (see, for instance, [8]). On the other hand, if the calculus is H ∞ -bounded then the operator A is dissipative in some equivalent Hilbert space inner product (refer to [8] for details). In sum, McEnnis' result can be achieved under the weaker conditions that combine sectoriality (of the spectrum) with bounded H ∞ -calculus. We apply the Naimark-Arocena Representation Theorem to obtain the Kreȋn space unitary dilation, though. So this note serves as a slightly more general result than the one by McEnnis as well as an alternate of its proof.
Apart from the present section, which functions as a brief introduction, this note comprises two other sections: Section 2 includes some preliminaries while Section 3 presents the results.

Preliminaries
In the sequel we assume that all Hilbert spaces are complex and separable. Given a Hilbert space (H, ·, · ), we denote by L(H) the linear space of all bounded linear operators on H. When A is a linear operator which is not everywhere defined on H, we write D(A) for its domain. The symbol R(A) stands for the range of A. The numerical range of A is the set ) are its resolvent set, spectrum and resolvent, respectively.
2.1. Holomorphic semigroups. References on semigroup theory abound; [11] is amongst the classical textbooks on the subject. We recall that a strongly continuous (one-parameter) semigroup on a Hilbert space (H, ·, · ) is a family {T (t)} ⊆ L(H) parameterized by t ≥ 0 that satisfies the following conditions: (i) T (0) = 1 and T (s + t) = T (s)T (t) for all s, t ≥ 0.
(ii) T (t) converges strongly to 1 as t → 0 + . When T (t) ≤ 1 for all t ≥ 0, {T (t)} is said to be a contraction semigroup.
The infinitesimal generator A of a strongly continuous semigroup {T (t)} is defined as Ax := lim t→0 + t −1 (T (t) − 1)x being the domain D(A) of A the set of those x ∈ H for which the limit exists. A is known to be a densely defined closed operator.
We also recall that a linear operator A on a Hilbert space Proposition 2.1. The following assertions are equivalent:  Let 0 < ω ≤ π 2 . An L(H)-valued function T on the additive semigroup is called a strongly continuous holomorphic semigroup if: (ii) T (0) = 1 and T (z + z ′ ) = T (z)T (z ′ ) for all z, z ′ ∈ X(ω).
Proposition 2.2. The following assertions are equivalent: The (m -) θ-accretive operators were extensively studied by T. Kato [9]. He called them (m -) sectorial operators but nowadays the term "sectorial" is referred to a different type of operator. In the following we will call A a sectorial operator if σ(A) ⊆ S 0,θ for some 0 < θ < π and, for each θ < φ < π, sup{ λR(λ, A) : λ ∈ C \ S 0,φ } < ∞. We will restrict ourselves to sectorial operators of angle 0 < θ < π 2 . A comprehensive account of sectorial operators is [8]. Therein the definition of sectorial operator is different from the one we adopt, though. Roughly, our corresponds with −A being sectorial in the above mentioned monograph.
The most famous result on the numerical range is the Toeplitz-Hausdorff Theorem which asserts that the numerical range of any (perhaps unbounded and not densely defined) linear operator on a complex or real (pre-)Hilbert space is convex. The convexity of the numerical range is one of the key elements in showing the following result from [9] (cf. [10]).
By combining Proposition 2.3 with Proposition 2.2 we get that, under the assumption that . Hence, if we consider only real t, we obtain that Holomorphic semigroups are somehow in between the general class of strongly continuous semigroups and the particular class of uniformly continuous semigroups (for which the infinitesimal generator is bounded). The functional calculus for sectorial operators gives a one-to-one correspondence between sectorial operators A with 0 ≤ θ < π 2 and bounded holomorphic strongly continuous semigroups {T (z)} on X( π 2 − θ) (see [8]). Therefore (2.3) is granted under the weaker condition of sectoriality. On the other hand, We replace (2.1) by the following hypothesis: there exist β ≥ 0 such that A − β is sectorial of angle 0 < θ < π 2 and Re Ax, x 0 ≤ β x 2 0 for all x ∈ D(A), with ·, · 0 an equivalent scalar product on H. This happens, for instance, if A − β is sectorial and has bounded H ∞ -calculus with H ∞ -angle less than π 2 (the reader is referred to [8] for the definitions). We remark that it is not always the case that a sectorial operator A on a Hilbert space, with sectorial angle less than π 2 , has bounded H ∞ -calculus.

Kreȋn spaces.
As familiarity with operator theory on Kreȋn spaces is presumed, only some notation is introduced. We emphasize that the common Hilbert space notation is carried over into the Kreȋn space setting.
Given a fundamental decomposition K = K + ⊕ K − of the Kreȋn space (K, ·, · ), we write |K| for K viewed as the Hilbert space relative to the fundamental decomposition. Therefore, if J is the corresponding fundamental symmetry, that is, By L(K) we mean the space of all everywhere defined continuous linear operators on the Kreȋn space K. The space L(K) has the structure of a Banach space depending on the choice of a fundamental decomposition and the associated Hilbert space |K|. The corresponding operator norm for L(K) is the norm · of L(|K|). We point up that any two operator norms for L(K) are equivalent and provide its topology. If K and G are two Kreȋn spaces, L(K, G) and the operator norm are defined likewise.
For each A ∈ L(K, G) there is a unique A * ∈ L(G, K) so that Ax, y G = x, A * y K for all x ∈ K and y ∈ G.
We say that (i) P ∈ L(K) is a projection if The regular subspaces of K are those that are the ranges of projections. If F is a regular subspace of K we write P F to indicate the orthogonal projection from K onto F.
Standard references on Kreȋn spaces and operators on them are [2], [3] and [4]. We also refer to [6] and [7] as authoritative accounts of the subject.
for all h : Γ → H with finite support and all ξ ∈ Γ.
Then there exist a Kreȋn space (K, ·, · K ) containing (H, ·, · ) as regular subspace and a unitary representation U (s) of Γ in K such that: Conversely, if there exist a Kreȋn space (K, ·, · K ) containing (H, ·, · ) as regular subspace and a unitary representation U (s) of Γ in K such that 1) and 2) hold, then there exists a kernel k satisfying (i), (ii), (iii) and (iv).

The Dilation
Hereafter {T (t)} is a strongly continuous semigroup of bounded linear operators on a Hilbert space (H, ·, · ) with infinitesimal generator A.
Let ·, · A be the graph scalar product on D(A): x, y A := x, y + Ax, Ay (x, y ∈ D(A)).
By the Riesz-Fréchet Representation Theorem, there exists a bounded linear operator R on (D(A), ·, · A ) such that Ax, y + x, Ay = x, Ry A = Rx, y A for all x, y ∈ D(A).
Since A is a densely defined closed operator on (H, ·, · ), the von Neumann Theorem (see [12]) assures that A * A, AA * are selfadjoint and 1 + A * A, 1 + AA * are invertible. Let h ∈ H and write h = (1 + A * A)y, y ∈ D(A). For all x ∈ D(A), If  Proof.
On the other hand, B is m-dissipative, for B is sectorial of angle 0 < θ < π 2 and dissipative. Therefore, B generates a strongly continuous bounded holomorphic semigroup {S(z)} on X( π 2 −θ) such that S(t) ≤ 1 in the real semi-axis t ≥ 0. As T (t) = e βt S(t) for all t ≥ 0, the result follows.
The arguments applied to A can be reproduced on the infinitesimal generator A * of the adjoint semigroup {T (t) * } to obtain a symmetric operator H * on (H, ·, · ) with D(H * ) = D(A * ) such that
We get two densely defined symmetric sesquilinear forms bounded from above with upper bound 2β by setting Both forms are closable [9]. Their corresponding closures g[·, ·] and g * [·, ·] are densely defined closed symmetric sesquilinear forms such that g[x, x] ≤ 2β x 2 for all x ∈ D(g) and g * [u, u] ≤ 2β u 2 for all u ∈ D(g * ) [9]. The Friedrichs Theorem for symmetric forms (cf. [9]) gives two selfadjoint operators G, G * which are bounded from above by 2β and satisfy A relevant fact established in [9] yields G = G * . Consider the polar decomposition G = J|G|, where J is a selfadjoint partial isometry and |G| is selfadjoint and nonnegative with D(|G|) = D(G). Then write The proof is omitted. Now we are ready to apply the Naimark-Arocena Representation Theorem to obtain a unitary dilation of {T (t)}. First we suppose that the infinitesimal generator A is as in Lemma 3.1. To comply with the hypotheses of the theorem, take Γ = R and define f : It follows that f is Hermitian and f (0) = 1. Hence the kernel (s, t) → f (s − t) is an operator-valued Toeplitz kernel on R.
If v(= v h ) is given by Therefore, by (3.3),
We first consider the case ξ > 0. The difference between the expressions for S(h, ξ) and S(h, 0) depends on the set supp(h) ∩ [−ξ, 0). Particular case: We apply Lemma 3.3 (c) and get We already mentioned that the cases to be considered are linked up with the points in supp(h) ∩ [−ξ, 0). Next we will see that all cases can be reduced to the particular one. If . Therefore, h −σ1 and ξ + σ 1 in the sum S(h −σ1 , ξ + σ 1 ) are like in the particular case. Whence Since h and −σ 1 in the sum S(h, −σ 1 ) accord with the particular case as well, we get S(h, −σ 1 ) ≤ e −8βσ1 S(h, 0), hence (3.7). The case σ 3 ≤ −ξ < σ 2 can be carried over into the previous one and so on.