On m-Isometric Semigroups, and 2-Isometric Cogenerators

It is known that a C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_0$$\end{document}-semigroup of Hilbert space operators is m-isometric if and only if its generator satisfies a certain condition, which we choose to call m-skew-symmetry. This paper contains two main results: We provide a Lumer–Phillips type characterization of generators of m-isometric semigroups. This is based on the simple observation that m-isometric semigroups are quasicontractive. We also characterize cogenerators of 2-isometric semigroups. To this end, our main strategy is to construct a functional model for 2-isometric semigroups with analytic cogenerators. The functional model yields numerous simple examples of non-unitary 2-isometric semigroups, but also allows for the construction of a closed, densely defined, 2-skew-symmetric operator which is not a semigroup generator.


Introduction
Throughout this paper, we let H and E denote complex Hilbert spaces. On their own, these are assumed to be completely general, while together, we will typically let E denote a certain subspace of H.  (i) T − I is injective, and has dense range.
It is classical, e.g. [20,Chapter III,Section 8], that if T ∈ L is a contraction, and T − I is injective, then T is the cogenerator of a contractive C 0 -semigroup. Moreover, the semigroup is isometric whenever T is. Note that if T is isometric, then (1.2) is trivially satisfied for any w ≥ 0. Hence, Theorem 1.2 has a natural converse in the case where m = 1. We obtain a similar result for m = 2: Theorem 1.3. Let T ∈ L be a 2-isometry. Assume further that 1 / ∈ σ p (T ), and that there exists w ≥ 0 for which (1.2) is satisfied. Then T is the cogenerator of a C 0 -semigroup (T t ) t≥0 , which is 2-isometric, and quasicontractive with parameter w.
Note that Theorem 1.3 does not require T − I to have dense range. This is much like in the isometric case: If T is an isometry, then T and T * have the same invariant vectors, so T − I has dense range if and only if it is injective. Our results imply that T − I has dense range under the more general hypothesis of Theorem 1.3, but the author has not found a direct proof.
The proof of Theorem 1.3 relies primarily on two non-trivial results about 2-isometries, namely a Wold-type decomposition theorem, and a functional model for analytic 2-isometries.
The first of these results (Theorem 2.10) is due to Shimorin [19], and states that any 2-isometry T can be written as a direct sum of a unitary operator, and an analytic 2-isometry. By some classical properties of unitary cogenerators, this allows us to reduce the proof of Theorem 1.3 to the case where T is analytic.
The second result (Theorem 2.11) states that if T ∈ L = L(H) is an analytic 2-isometry, and E = H T H, then T is unitarily equivalent to the operator M z , multiplication by the identity function z → z, acting on a harmonically weighted Dirichlet space D 2 μ (E) of E-valued analytic functions on D. The parameter μ is a measure on the unit circle T, with values in L + = L + (E). The correspondence between T and μ is essentially bijective. Theorem 2.11 was proved by Richter [15] in the case where dim E = 1, and extended to the general case by Olofsson [14]. The main idea behind Theorem 1.3 is that an analytic 2-isometry T ∈ L is the cogenerator of a C 0 -semigroup if and only if M z is the cogenerator of a C 0 -semigroup on the corresponding space D 2 μ (E). Such a semigroup is necessarily given by the multiplication operators (M φt ) t≥0 , where φ t : z → exp(t(z + 1)/(z − 1)). Our strategy is then to determine all μ such that the operators (M φt ) t≥0 form a C 0 -semigroup on D 2 μ (E). After reduction to the analytic case, Theorem 1.3 follows from: |1−ζ| 2 is an L + -valued measure, and there exists w 2 ≥ 0 such that On the other hand, if μ is any measure satisfying the above inequality, then |1−ζ| 2 defines an L + -valued measure. Hence, when attempting to verify condition (iii) of Theorem 1.4, the inequality (1.5) is a natural first step.
Any bounded operator A generates an invertible C 0 -semigroup (e tA ) t≥0 . As a special case of [5, Theorem 2.1] (or Theorem 1.1), stated explicitly as [5,Corollary 2.3], the semigroup is m-isometric if and only if A is m-skewsymmetric. A limitation of this conclusion is that if m is even, then by [2, Proposition 1.23], (e tA ) t≥0 is in fact (m − 1)-isometric. In particular, any 2-isometric C 0 -semigroup with bounded generator is unitary. On the other hand, Theorem 1.4 allows one to produce numerous examples of non-unitary 2-isometric semigroups. In particular, such semigroups exist. One may also use Theorem 1.4 to construct a closed, densely defined, 2-skew-symmetric operator A, with the property that λ − A : D(A) → H is surjective for any λ > 0, but which is not the generator of a C 0 -semigroup. This shows that the conditions (1.1) and (1.2) are not superfluous.
The paper is organized as follows: In Sect. 2 we introduce some notation and preliminary material. In Sect. 3 we discuss m-isometric semigroups. In particular, we prove Theorems 1.

Notation and Preliminaries
We use the notation D := {z ∈ C; |z| < 1} for the open unit disc, and T := {z ∈ C; |z| = 1} for the unit circle of the complex plane C. By λ we denote Lebesgue (arc length) measure on T, while dA will signify integration with respect to area measure on C. We also use ∂Ω and Ω to denote the boundary and closure of Ω ⊂ C, respectively. Given integers m ≥ k ≥ 0, we let m k = m! k!(m−k)! denote the standard binomial coefficients. If the integers m and k do not satisfy the prescribed inequalities, then we set m k = 0. With this convention, the well-known relation is valid whenever m ≥ 1 and k ∈ Z. This will be used repeatedly.
Given an operator A, we let σ(A), σ p (A), σ ap (A), and W (A) respectively denote the spectrum, point spectrum, approximate point spectrum, and numerical range of A, i.e. Given a family {S i } i∈I of subsets of H, we let i∈I S i denote the smallest closed subspace of H that contains each S i .

m-Isometries, and m-Skew-Symmetries
Let m ∈ Z ≥0 , T ∈ L, and define Apart from a normalizing factor, this agrees with the notation from [2]. A straightforward consequence of (2.1) is that We also have the following formula, valid for k ∈ Z ≥0 , T ∈ L: Note that with our notational convention for binomial coefficients, the above right-hand side has at most k + 1 non-zero terms. If β m (T ) = 0, then we say that T is an m-isometry. By (2.2), any such operator is also an (m + 1)-isometry. Moreover, (2.3) implies Gelfand's formula for the spectral radius yields that σ(T ) ⊆ D. A more careful analysis reveals that σ ap (T ) ⊆ T, see [2, Lemma 1.21]. By the general fact that ∂σ(T ) ⊆ σ ap (T ), it follows that an m-isometry is either invertible, in which case σ(T ) ⊆ T, or it is not invertible, in which case σ(T ) = D.
Given m ∈ Z ≥1 , and an operator A, we define the sesquilinear form α A m by We say that A is m-skew-symmetric if α A m vanishes identically. The relation between m-isometries and m-skew-symmetries has been frequently exploited in previous works [5,11]. For easy reference, we state and prove the following result, which is implicit in [17, p.
Proof. Since It is clear that (2.7) holds for m = 0. For a general m, we use (2.2) to see that Since β m (T ) * = β m (T ), the above identity, with x 1 and x 2 interchanged, implies On m-Isometric Semigroups, and 2-Isometric Page 7 of 30 27 Adding these two identities, Since T + I = 2A(A − I) −1 , and T − I = 2(A − I) −1 , we therefore have Assuming that (2.7) holds, (2.6) implies Hence, we obtain (2.7) by induction over m.

C 0 -Semigroups
By a semigroup we mean a one-parameter family (T t ) t≥0 ⊂ L, such that T 0 = I, and T s+t = T s T t for s, t ≥ 0. For a detailed treatment of the facts outlined below, we refer to [8,Chapter II].
A semigroup is called a C 0 -semigroup, or strongly continuous, if for every If (2.8) holds with M = 1, then we say that (T t ) t≥0 is quasicontractive with parameter w. A quasicontractive semigroup with parameter 0 is simply called contractive.
The (infinitesimal) generator of (T t ) t≥0 is the operator A defined by

Its domain D(A)
is the subspace of y ∈ H such that the above limit exists. The generator of a C 0 -semigroup is closed, densely defined, and uniquely determines the semigroup. If y ∈ D(A) and t ≥ 0, then T t y ∈ D(A), and It is easy to show that in this sense, Not every closed, densely defined operator is the generator of a C 0semigroups. A fundamental result in this direction is the so-called Lumer-Phillips theorem: Theorem 2.2. Let w ≥ 0, and A be an operator. Then A is the generator of a quasicontractive C 0 -semigroup with parameter w if and only if the following conditions hold: (i) A is closed and densely defined.
The above theorem is typically stated for w = 0. The general version follows if we consider the operator A − w, and the corresponding semigroup (e −wt T t ) t≥0 . An elementary calculation using the inner product shows that A is 0dissipative if and only if Consequently, λ − A is injective in this case. Moreover, the above inequality turns out to be the appropriate analogue of (ii) when studying semigroups of operators on Banach spaces.
If the above conditions (i)-(iii) hold, then λ − A : D(A) → H is in fact surjective for any λ > w.

Operator Measures
Let S denote the Borel σ-algebra of subsets of T. An L + -valued measure is a finitely additive set function μ : S → L + with the property that for every x, y ∈ E, the set function μ x,y : E → μ(E)x, y defines a complex regular Borel measure. For each E ∈ S, it holds that (2.10) We refer to the proof of [13, Proposition 1.1]. Given a bounded (Borel) measurable function f : T → C, we can define the sesquilinear form J f : (x, y) → T f dμ x,y . It follows from (2.10), that By standard functional analytic considerations, the above inequality implies the existence of a uniquely determined operator I f ∈ L such that I f x, y = T f dμ x,y . We denote the operator I f by T f dμ. The integral thus defined satisfies the triangle type inequality We will need the following version of the Cauchy-Schwarz inequality: (2.11) Proof. When f and g are simple functions, (2.11) follows from (2.10), and the Cauchy-Schwarz inequality for finite sums. General functions are approximated in the standard fashion. Two instances of the above integral will be particularly interesting to us, namely the Fourier coefficientŝ and the Poisson extension If we let r = |z|, then By the Weierstrass test, this series converges uniformly in ζ. Using term by term integration, we conclude that With the above construction, the integral f dμ is only defined when f is bounded. As a remedy for this, adequate for our purposes, we use the following construction: Let μ be an L + -valued measure, and h a scalar-valued function. If there exists C > 0 such that then one can define a new set function μ h by By Lemma 2.4, the above right-hand side has modulus less than This estimate implies that μ h is another L + -valued measure. If f is bounded, then we may take f dμ h as a definition of fh dμ.
Remark 2.5. We will only use the above construction of fh dμ in the setting where h is a fixed function. However, the following may be of independent interest: If f 1 h 1 = f 2 h 2 , and the measures μ i = μ hi are defined as above, then Hence, Let f, g : T → E be continuous functions, and identify these with their respective Poisson extensions. For 0 < r < 1 and ζ ∈ T, and the right-hand side converges uniformly in ζ. Integrating with respect to dλ(ζ) yields This motivates us to define dμ ·, · by provided that the above right-hand side is absolutely convergent. It seems clear that any reasonable definition of dμ ·, · should satisfy (2.12). On the other hand, it is a bit awkward to require so much regularity for a function to be square integrable. The next example is a digression from the primary topic of this paper, but may still be of interest.
where (e k ) ∞ k=1 is some orthonormal sequence in E. Then μ is an L + -valued measure. For a simple function f = n x n 1 En , it might seem reasonable to define dμ f, f as the finite sum However, if f N = N k=1 e k 1 I k , then the above sum is equal to N , even though f N L ∞ = 1. This may help explain why we have chosen (2.12) as our definition of dμ ·, · .

Function Spaces
The space of analytic polynomials N k=0 a k z k with coefficients a k ∈ E is denoted by P a (E). As a notational convention, we write P a in place of P a (C). The same principle applies to all function spaces described below.
Let f be a function which is analytic in a neighbourhood of the origin. The kth Maclaurin coefficient of f is denoted byf (k). By D a (E), we denote the space of E-valued analytic functions whose Maclaurin coefficients (f (k)) ∞ k=0 decay faster than any power of k. These are precisely the analytic functions on D which extend to smooth functions on D. The If f ∈ H 2 (E), then the radial boundary value f (ζ) := lim r→1 − f (rζ) exists for λ-a.e. ζ ∈ T. The radial boundary value function satisfies (2.14) In the case where E = C, these facts will be included in any reasonable introduction to Hardy spaces. For general E, we refer to [12, Chapter III], or [16,Chapter 4].
Given an L + -valued measure μ, and an analytic function f : D → E, we define the corresponding Dirichlet integral The integrand is non-negative, so the integral is well-defined. For z = ρζ, ζ ∈ T, the power series expansions of P μ and f yield Integrating this identity with respect to ρ dρ dλ(ζ), over 0 < ρ < r < 1, and ζ ∈ T, gives us the formula 1 By monotone convergence, If the resulting series is absolutely convergent (say if f ∈ D a (E)), then we may of course replace the lim r→1 − with an evaluation at r = 1. We define the harmonically weighted Dirichlet space D 2 μ (E) as the space μ (E), then we define the sesquilinear Dirichlet integral which is finite by the Cauchy-Schwarz inequality.
Proof. By (2.14), M z : H 2 (E) → H 2 (E) is an isometry. Hence, The above right-hand side equals 1 2π T dμ f, f by definition. Given an analytic function f : D → C, and ζ ∈ T, we define the corresponding local Dirichlet integral This is a convenient shorthand for the Dirichlet integral D δ ζ (f ), where δ ζ denotes a (scalar) unital point mass at ζ. If μ is a positive scalar-valued measure, then it is immediate from Fubini's theorem that A useful tool for calculating local Dirichlet integrals is the so-called local Douglas formula. The proof, and a slightly more general version of the statement, can be found in [7, Chapter 7.2]: Recall that a function θ ∈ H 2 is called inner if |θ(ζ)| = 1 for λ-a.e. ζ ∈ T. If θ is inner and f ∈ D a , then  In the general case of L + -valued measures, the concept of a local Dirichlet integral does not appear to be well studied. We derive the following substitute for (2.17). Lemma 2.9. Let μ be an L + -valued measure, and x, y ∈ E. If f ∈ D 2 μx,x and g ∈ D 2 μy,y , then the integral is absolutely convergent, with |D μx,y (f, g)| ≤ f D 2 μx,x g D 2 μy,y . In particular, Proof. We need to show that The inner integral equals |f (z)g (z)|P |μx,y| (z). By an application of Lemma 2.4, we obtain The remainder of the statement follows from the Cauchy-Schwarz inequality, and Fubini's theorem.

Analytic Operators
An operator T ∈ L is called analytic if ∩ n≥0 T n H = {0}. It is clear that an analytic operator cannot have any non-zero eigenvalues. In particular, an analytic m-isometry does not have eigenvalues, since σ ap (T ) ⊆ T.
A good reason to study analytic 2-isometries is the existence of a socalled Wold decomposition: For T ∈ L, we let E = H T H. The dimension of E is called the multiplicity of T . Furthermore, define the spaces H u = ∩ n≥0 T n H, and H a = n≥0 T n E. The following is a special case of [19,  The class of analytic 2-isometries with multiplicity 1 can be completely described in terms of M z , multiplication by the function z → z, acting on harmonically weighted Dirichlet spaces [15]. This result was later generalized to arbitrary multiplicity by Olofsson [14]: The above correspondence is essentially one-to-one; the operators

m-Isometric Semigroups
By an m-isometric semigroup we mean a C 0 -semigroup (T t ) t≥0 such that each T t is an m-isometry. In the transition from individual operators to C 0semigroups, the following two lemmas are useful: y1,y2 is given by where α A m is defined by (2.5).
In particular, f Proof. This is trivial for by the product rule. By induction one obtains provided that y 1 , y 2 ∈ D(A m ). This is equivalent to (3.1).
Proof. Use the fundamental theorem of calculus, (2.1), and induction over m.
As a first application of these identities, we prove a semigroup analogue of (2.4). This is more precise than condition (ii) of [  Proof. For y ∈ D(A j ), let f (t) = T t y 2 . By Lemma 3.1, f (j) (t) = α A j (T t y). Moreover, this is a continuous function, and by Lemma 3.2, This identity implies By (2.4), it holds for any k ∈ Z ≥0 that If y ∈ D(A m−1 ), then we may let k → ∞, in order to obtain We want to extend the above identity to y ∈ H. For this it is sufficient to prove that each α A j is bounded. Given m distinct times (t k ) m−1 k=0 , the numbers ( T t k y 2 ) m−1 k=0 uniquely determine α A j (y). By linear algebra, it even holds that where the numbers (a jk ) do not depend on y. It follows that each α A j is a bounded quadratic form.
Together with (2.9), Theorem 3.3 implies the well-known result that if A is the generator of an m-isometric semigroup, then σ(A) ⊆ {z ∈ C; Re z ≤ 0}. A novel result is the following:

Corollary 3.4. Let (T t ) t≥0 be an m-isometric semigroup with generator A.
Then, there exists a ∈ R, b ∈ (a, ∞), such that Proof. By Theorem 3.3, α A 1 (y) = 2 Re Ay, y defines a bounded quadratic form on H.
Proof. Since α A 0 = I, Theorem 3.3 yields that for some polynomial p. It is clear that the right-hand side is dominated by e wt for some w ≥ 0.

Proof of Theorem 1.1
Suppose that A is the generator of an m-isometric semigroup (T t ) t≥0 . Property (i) holds for any generator. By Corollary 3.5, (T t ) t≥0 is quasicontractive for some parameter w ≥ 0. Properties (ii) and (iii) are implied by the Lumer-Phillips theorem (Theorem 2.2). Since each T t is m-isometric, (3.3) implies that α A m vanishes, i.e. A is m-skew-symmetric. This proves that the conditions (i)-(iv) are necessary.
Suppose on the other hand that A satisfies (i)-(iv). The first three conditions, together with the Lumer-Phillips theorem, imply that A generates a C 0 -semigroup (T t ) t≥0 , quasicontractive with parameter w ≥ 0. The assumption that A is m-skew-symmetric implies that each T t is m-isometric, by (3.2).

2-Isometric Cogenerators
The purpose of this section is to prove Theorem 1.3, which essentially amounts to proving Theorem 1.4.
We recall the hypothesis of Theorem 1.3: T ∈ L is a 2-isometry, T − I is injective, and there exists w ≥ 0 such that (4.1) The assertion that we wish to prove is that T is the cogenerator of a C 0semigroup (T t ) t≥0 , and that this is quasicontractive with parameter w. The first step is a reduction to the case of analytic operators. By the Wold-decomposition (Theorem 2.10), T = T u ⊕ T a , where T u is unitary and T a is analytic. Since 1 / ∈ σ p (T ), we know that 1 / ∈ σ p (T u ). By [20, Chapter III, Section 8], T u is the cogenerator of a C 0 -semigroup of unitary operators on H u . This is clearly quasicontractive for any w ≥ 0. It therefore suffices to show that T a is the cogenerator of a quasicontractive C 0 -semigroup on H a .
By orthogonality and invariance of the subspaces H u , H a , and the fact that T u is unitary, Theorem 2.10 further implies that (4.1) is equivalent to By Proposition 2.6, it is sufficient to verify this for f ∈ P a (E), and by Proposition 2.7, T satisfies (4.1) if and only if If we for a moment assume the validity of Theorem 1.4, then the above condition implies that (M φt ) t≥0 is a C 0 -semigroup on D 2 μ (E), and that this is quasicontractive with parameter w. Let (T t ) t≥0 = (V * M φt V ) t≥0 . This defines a C 0 -semigroup on H, it's quasicontractive with parameter w, and its cogenerator is given by V * M z V = T .

Proof of Theorem 1.4
Let us recall the statement of Theorem 1.4: If μ is an L + -valued measure on T, then the following are equivalent: (i) For every t ≥ 0, M φt ∈ L(D 2 μ (E)), and the family (M φt ) t≥0 is a C 0semigroup.
(ii) There exists w 1 ≥ 0 such that (iii) The set functionμ : |1−ζ| 2 is an L + -valued measure, and there exists w 2 ≥ 0 such that If one (hence all) of the above conditions is satisfied, then the C 0 -semigroup (M φt ) t≥0 is 2-isometric, has cogenerator M z , and is quasicontractive for some parameter w ≥ 0. Moreover, the optimal (smallest) values of w, w 1 , and w 2 coincide. In the case E = C, there is a quite direct proof that (i)⇔(iii). In the general case, we essentially use the same ideas, although they are somewhat obscured by technicalities. For this reason, we begin with a preliminary discussion of the case E = C. In general, the qualitative assertions that (M φt ) t≥0 is 2-isometric and quasicontractive are fairly immediate from (i). The argument also shows that (i)⇒(ii). The implications (ii)⇒ (iii) ⇒(i) require a bit more work. It will be evident that the optimal values of w, w 1 , and w 2 coincide.
By (2.18) and this holds for μ-a.e. ζ ∈ T. Using Fubini's theorem (2.17), the above formula implies that Adding From this, we conclude that M φt : For the verification that, under such circumstances, (M φt ) t≥0 is indeed a C 0 -semigroup, we refer to the general case below.
Remark 4.1. The above argument shows that for E = C, condition (i) may be weakened to: (i ) There exists t > 0 such that M φt ∈ L(D 2 μ ). The author has not found a proof that the same phenomenon occurs in the general case.
A simple calculation shows that M z is the cogenerator of (M φt ) t≥0 . M z is 2-isometric, and by Lemma 2.1, the corresponding generator is 2skew-symmetric. By Theorem 1.1, (M φt ) t≥0 is 2-isometric. By Corollary 3.5, (M φt ) t≥0 is quasicontractive for some w ≥ 0. For any such w, Theorem 1.2 implies that In particular, the above inequality is satisfied for f ∈ P a (E). By Proposition 2.7, condition (ii) holds with w 1 = w.
We now prove (4.6). By Lemma 4.3, the right-hand side is well-defined. For f ∈ P a (E), let {e n } ⊂ E be an orthonormal basis of a finite-dimensional subspace containing the range of f . Then f = n f n e n , where each f n ∈ P a . Defining μ m,n = μ em,en , a calculation shows that For |μ m,n |-a.e. ζ ∈ T, Note that By Lemmas 2.4, and 4.3, the above right-hand side is |μ m,n |-integrable. Hence, by dominated convergence, This proves that if f ∈ P a (E), and μ satisfies (4.3), then (4.6) holds. The proof of (4.7) is similar. Reusing the above notation, By Lemma 2.9, each one of these Dirichlet integrals can be computed as By polarization, and Lemma 4.2, for |μ m,n |-a.e ζ ∈ T. Moreover, By Lemmas 2.4, and 4.3, the right-hand side is |μ m,n |-integrable. By dominated convergence, This shows that (4.7) holds whenever f ∈ P a (E), and μ satisfies (4.3). The proof that (ii) ⇒ (iii) is complete. (iii) ⇒ (i) : We are assuming the existence of w 2 ≥ 0, such that Recall that φ t : z → exp(t(z + 1)/(z − 1)). The core of our proof is the following formula: Lemma 4.4. Let μ be an L + -valued measure that satisfies (4.9). If f ∈ P a (E), and t > 0, then Step 1, (i = 0) : Using (2.1), Step 2, (recursion formula) : Using (2.1) again, Step 3, (induction) : By step 1, it holds that LHS where (α A j ) m−1 j=0 is given by (2.5).
A significant difference from the m-isometric case is that we have no reason to expect the forms (α A j ) m−1 j=0 to be bounded. Therefore, we obtain no evidence that m-concave semigroups are quasicontractive by necessity. On the other hand, from Proposition 6.2, we have that Each form x → β j (T 1 )x, x is bounded on H. Together with the semigroup property, this implies that T t 2 (1 + t) m−1 . From (2.9), we therefore obtain:  I am also indebted to the anonymous referee for carefully reading and commenting on several versions of this manuscript. In particular, the suggestion to include a more thorough discussion on generators led to substantial improvements not only of the presentation, but also of the results.
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