Domination for subordinated positive semigroups


 We first give a necessary and sufficient condition for -f(B) to be dominated by -f(A), where f is a completely Bernstein function, B and A are C0-semigroup generators. Then we prove that there is no domination relationship between a semigroup and the subordinated semigroup if additional conditions are satisfied.


Introduction
f (A) = f 0 (A). It was proved by [15,Theorem 4.1] that f (A) is a sectorial operator with spectral angle ω(f (A)) = ω(A). This kind of function calculus is sometimes called Hirsch function calculus which has many applications and generalizations, for example, A. Gumilko et al( [13], [14] and [16]) use this approach to investigate the approximation theory of semigroups and the ergodicity properties of subordinated semigoups; C. Batty et al( [9]) extend the domain of this functional calculus to function space consisting of N P + functions and several resolvent representations are given here; the generalization of function calculus in Besov space and some related topics are dicussed by C. Batty et al( [10] and [11]).
In this paper, we first investigate the domination relationship between Then we prove that there is no domination relationship between positive C 0 -semigroup To be more precise, we first prove that the irreduciblity and mean-ergodicity are preserved by Hirsch functional calculus, then use the result proved by W. Arendt([6, and thus f (x) = x by uniqueness of functional calculus and semigroup generator.

Preliminaries
2.1. Basic Notations. Throughout this paper, X is a Banach space, E is a Banach lattice, L(X), L(E) are the Banach algebras of all bounded linear operators on X, E. We always assume that A is a closed unbounded operator and densely defined on X(or E). We will denote by D(A), N (A), R(A) the domain, kernal and range of A respectively. And by ρ(A), σ(A) we denote the resolvent set and spectrum of A. R(λ, A) := (λ − A) −1 means the resolvent of A at λ if λ ∈ ρ(A). A sector Σ θ is defined as Σ θ := {λ ∈ C|λ = 0, | arg λ| < θ} for θ ∈ (0, π) and Σ 0 = (0, ∞). We write H ∞ (Σ θ ) for the space of all holomorphic function on the open set Σ θ with super norm f H ∞ (Σ θ ) := sup x∈Σ θ f (x) , and H ∞ 0 (Σ θ ) for all holomorphic function on the open set Σ θ with polynomial limit,

Sectorial Operator and Functional
Calculus. Sectorial operator is widely used in operator theory, it has many equivalent definitions, in this paper we use the following one.
Now we list the definition of functional calculus, the elaborate details, properties and proofs can be found in [15], [17] and [24].
and A is a sectorial operator of angle ω < φ, then Definition 2.4. Let A be a sectorial operator of angel ω, f be a holomorphic function on Σ φ , ω < φ. If there exist a function e ∈ ξ(Σ φ ) such that ef ∈ ξ(Σ φ ) and 0 ∈ ρ(e(A)), then Since Laplace transform is used in resolvent theory frequently, the following special Laplace inversion formula is quiet useful.
More precisely, g is given by where Γ is the boundary of sector Σ ω 1 oriented in the positive sense and ω 1 ∈ ( π 2 , φ) is arbitrary.
We use BF to denote the space of Bernstein function.
where a, b > 0 and m(t) is the complete monotone density of µ.
We use CBF to denote the space of completely Bernstein function.

Proposition 2.10. A completely Bernstein function f has an analytic representation in upper plane which is given by
The following functional calculus definition for Bernstein function can be found in many references such as [9], [15] and [23].
The operator f (A) is called the Bernstein function of A. If {µ t } is the vaguely continuous convulution semigroup corresponding to f ,

Domination for Subordinated Semigroups
Here we give a brief introduction on ordered Banach spaces, positive operators and resolvent positive operators. More details about these topics can be found in [4], [5] and [18].
We say that E has order continuous norm if each decreasing positive sequence in E converges.
Through out this paper, we always assume a Banach lattice E is generating with normal cone.
Definition 3.2. A vector-valued function f : (0, ∞) → E is said to be positive if f (λ) ∈ E + and will be denote by f ≥ 0.
The following vector-valued Bernstein theorem gives a nesessary and suffcient condition for completely monotonic function on ordered Banach space with order continuous norm.
We need the following lemma which involves the conception of integrated semigroups, we refer to [5] for more defintions and properties. Since we know that if −A generates a bounded positive C 0 -semigroup, then −A is resolvent positive, and thus −f (A) is resolvent positive for every f ∈ BF. Next lemma shows that if f ∈ CBF, then this conclusion is also hold withourt the generation of semigroup.
Since we can choose λ > 0 be a certain number, so we write g λ (z) as g(z) for convenience. If g(z) = 1 or g(z) = 1 1+z , the conclusion is obviously ture. If g ∈ H ∞ 0 (Σ ω ), let π 2 < φ < ω, by [17, Lemma 2. 3. 1] we know that, Since −A is a densely defined, resolvent positive operator, by lemma 3.5 we know that −A generates a once-integrated semigroup {T 1 (t)} such that, Since g(z) is completely monotonic, by using lemma 3.3 we know there exists an increasing function Φ(s) such that, And since g ∈ H ∞ 0 (Σ ω ), so by using lemma 2.5 we know there exists a function φ ∈ L 1 (0, ∞) such that, Combine these results together we have dΦ(s) = φ(s)ds, so φ(s) is non-negative, and Since once-integrated semigroup T 1 (t) is non-decreasing, this end the proof. A problem related to positive semigroup is: How to determin the order raletion of two semigroups on σ-order complete Banach lattice? For such a problem, W.Arendt gives the following result.
(2) The following inequality holds, where sign(f ) is an unique bounded operator on E satisfies: The inequality (3.1) also be called the abstract version of Kato's inequality. Therefore, a nature question is: The following theorem answers this question.

t)} is dominated by {T −A (t)} if and only if for every f ∈ CBF, {T −f (B) (t)} is dominated by {T −f (A) (t)}.
Proof. By using lemma 3.8 we only need to prove that: if the following inequality holds, Because E is a σ-complete order Banach lattice, if the preceding inequality holds in an operator core of f (B) and f (A) * , then it is hold for u ∈ D(f (−B)) and φ ∈ D(f (−A) * ). Now assume inequality (3.2) holds. Then we know that So we only need to prove that

So our goal is inequality
holds for every λ > 0. Since A is a sectorial operator, ω(A) = π 2 , and −A is resolvent positive, define a function G(x): This is a completely Bernstein function, then by using lemma 3.
H(x) obviously be a completely monotonic function, and H(0) = 1 µ , then by Bernstein theorem there exists a increasing function Φ(t) such that: Since H(0) = 1 µ = 0, then This complete the proof.
One purpose of Kato' inequality is to describle the positivity of semigroup {T −A (t)} by using the inequality that the generator −A satisfies. In [19], R. Nagel conjectured that some abstract version of Kato' inequality is equivalent to positivity of semigroups, one form of this equivalenence was proved in [2, Theorem 1.6].
In this characterization, the author use the conception 'sub-eigenvector set'. The element u is the sub-eigenvector of the operator A if u > 0, u ∈ D(A) and there exists a constant M ∈ R such that Au < M u.
So we naturally give the following two propositions. We need to prove that there exists a constant K ∈ R such that Since −A is resolvent positive, for every λ > M + 1, we have Proposition 3.12. Let A be a closed sectorial operator and −A is resolvent positive, Proof. Similar to the proof of theorem 3.9, we only need to prove that for every λ > 0, Since −A(λ + A) −1 is a bounded resolvent positive operator and generates a positive C 0semigroup. Then by weak Kato' inequality([2, Proposition 1.1]) we can easily prove that the target inequality hold.
Phillips showed that the Bernstein function of a bounded C 0 -semigroup generator still generates a bounded C 0 -semigroup. Because we do not assume that −A is a semigroup generator in proposition 3.11 and proposition 3.12. Therefore, quoting proposition 3.11, proposition 3.12 and [2, Theorem 1.6], we use a completely different method to prove that the completely Bernstein function of a bounded positive C 0 -semigroup generator still generates a bounded positive C 0 -semigroup.
If C 0 -semigroup {T (t)} is positive and bounded, then we introduce the concept of irreducibility of T (t).  Proof. By Definition 3.13, since {T −A (t)} is irreducible, assume 0 < x ∈ E and 0 < y ∈ E * , there exists a time t 0 ∈ (0, ∞) such that Then we need to prove that there exists a time t 1 ∈ (0, ∞) such that Let {µ t } be the vaguely continuous convolution semigroup corresponding to f , then Since y, T −A (t 0 )x > 0 and function f (·) := y, T −A (·)x : R + → R + is continuous, then there exist an interval U such that t 0 ∈ U and y, T −A (s)x > 0 for every s ∈ U . Then we only need to prove that there exists a time t 1 ∈ (0, ∞) such that This is equal to show that there exists a time t 1 ∈ (0, ∞) such that By [23, definition 5.4] we know that there exists a subordinator {S t } with state space [0, ∞) such that µ t (U ) = P (S t ∈ U ), then we need to show that exists a time t 1 ∈ (0, ∞) such that P (S t 1 ∈ U ) > 0, and this can be deduced from [12, Proposition 1.9(ii)] and assumption b > 0. Proof. If f ≡ 0, then by [ So for every x ∈ R(A), Ay = x, y ∈ X, we have By using preceding two lemmas, we can prove the following theorem, which assert that there is no ordered relationship between {T −A (t)} and {T −f (A) (t)} if some conditons are satisfied. do not hold for all t > 0. ( do not hold for all t > 0. Proof. In order to prove conclusion (1), we assume and 0 ≤ T −A (t) ≤ T −f (A) (t) hold for all t > 0.