Characterization of Orlicz admissibility

We extend two characterizations of admissible operators with respect to Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{L}^p$$\end{document} to more general Orlicz spaces. The equivalent conditions are given by the property that an associated operator generates a strongly continuous semigroup and in terms of a resolvent estimate.


Introduction
In this note we study operators C related to linear observation systems where A generates a C 0 -semigroup (T (t)) t≥0 on a Banach space X, the state space and x 0 ∈ X is the initial state.The operator C is the output operator (or "observation operator") mapping the domain of A (equipped with the graph norm) continuously into some Banach space Y , the output space.Since the (mild) solution x in (1) is given by the semigroup applied to the initial state, x(•) = T (•)x 0 , the output is formally given by y(•) = CT (•)x 0 .
Admissibility with respect to L p (or L p -admissibility) means that the mapping x 0 → CT (•)x 0 , initially defined only on D(A), has a continuous extension from X to L p (0, τ ; Y ).Here we distinguish between finite-time admissibility (or just admissibility), that is if τ < ∞ and infinite-time admissibility, that is if τ = ∞.
In the literature admissibility with respect to various spaces of output functions, different than L p , has been considered, see e.g.[9,13,29].In this introduction we focus on infinite-time admissibility, noting that all mentioned results can be formulated analogously for finite-time admissibility.Indeed, the infinite-time case is more interesting and the finite-time case can be derived by the infinite-time case since L p -admissibility is invariant under scaling of the semigroup and hence under shifting of the generator.The notion of admissibility is strongly related to the more general concept of well-posed linear systems being of central interest in system theory, see e.g.[26] for L 2 .
In this article we study two characterizations of this notion with respect to Orlicz spaces L Φ and E Φ ; an estimate on the resolvent of the Laplace transformed trajectory of the output y = CT (•) and an equivalence in terms of generation of a strongly continuous semigroup.These results extend earlier results for the special case of L p -admissibility due to Callier-Grabowski [6], Engel [3], as well as Le Merdy [21] and Haak [7].
Main result (c.f.Theorem 3.2).Let C : D(A) → Y be a bounded operator with respect to the graph norm of A. Then C is infinite-time E Φ -admissible if and only if the block operator matrix with domain generates a C 0 -semigroup on X ×E Φ (0, ∞; Y ), where Lx := 1 | [0,∞) Cx for x ∈ D(A).
A second characterization of L p -admissibility due to Le Merdy (p = 2) and Haak (p ≥ 1) relates to a conjecture originally formulated by Weiss in [28] (for p = 2 and, equivalently, for the dual problem of control operators).The p-Weiss conjecture states that infinite-time L p -admissibility of C is equivalent to the so-called infinitetime p-Weiss condition for C, i.e. sup Re z>0 a property which is easily seen to follow from L p -admissibility by Hölder inequality.The question thus is whether the p-Weiss condition is sufficient for L p -admissibility of C. Whereas the answer is negative in the general Banach space setting [28] (and p = 2), the problem has received much attention since then, with both positive results, as well as counterexamples.We mention here some of them and refer to [7] and [15] for detailed overviews.In [16,17] it is shown, that the 2-Weiss conjecture does not hold in arbitrary Hilbert spaces without further assumptions on the semigroup and the operators C. For Hilbert spaces the p = 2-case is known to hold true for exponentially stable, left-invertible semigroups, [28], as well as in the case of contraction semigroups and finite-dimensional output spaces, [14].For infinite-dimensional output spaces, the statement may fail even for semigroups of isometries [16].Le Merdy showed in [21] that the 2-Weiss conjecture holds true in the Hilbert space situation under the assumption of an analytic contractive semigroup.Moreover, he showed for Banach spaces and a bounded analytic semigroup that the 2-Weiss conjecture holds if and only if the operator (−A) 1 /2 defined via the holomorphic functional calculus is infinite-time L 2 -admissible.Haak [7] extended Le Merdy's results to more general p ≥ 1 as follows: If A generates a bounded analytic semigroup and A has dense range, then the p-Weiss conjecture holds if and only if (−A) 1 /p is infinite-time L p -admissible.He used generalized square function estimates for the operator A which are equivalent to (−A) 1 /p being infinite-time L p -admissible.
We continue these developments in the context of Orlicz spaces for a certain class of Young functions, which is called class P. Our second main result reads: Main result (c.f.Theorem 4.12).Suppose A generates a bounded analytic semigroup, the Young function Φ is of class P and either • A is a multiplication operator with σ(−A) ⊆ [0, ∞) or • Φ −1 is holomorphic on some sector containing the spectrum of −A and there exist constants m 0 , m 1 > 0 such that for z in this sector it holds that Then, for a bounded operator C : D(A) → Y the following are equivalent Here, Φ denotes the to Φ complementary Young function, see Section 2.
This paper is structured as follows.In Section 2 we recall the for us necessary definitions and facts on Young functions Φ and Orlicz spaces L Φ and E Φ .Furthermore, we introduce a subclass P of Young functions with polynomial behavior at 0 and ∞ which play a role when dealing with the functional calculus in the context of the Weiss conjecture.We shift more technical results on Orlicz spaces to the Appendix, where the reader can also find results on the dual space of vector valued Orlicz spaces, the shift semigroup on Orlicz spaces and a generalization of Minkowski's integral inequality to Orlicz spaces used in the ensuing sections.In Section 3 we formalize Orlicz admissibility for observation operators and prove Section 1, which is an adoption of the L p case by our previous investigations on the shift semigroup in Orlicz spaces.Section 4 is devoted to the extension of the Weiss conjecture for Orlicz spaces and the proof of Section 1. Furthermore, we give a sufficient condition for infinite-time L Φ -admissibility of Φ −1 (−A) in case of a semigroup on ℓ r , 1 ≤ r < ∞ generated by a multiplication operator.In Section 5 we discuss the duality concept of Orlicz admissible control and observation operators which allows to transfer all results on Orlicz admissibility to control operators.
In the following X and Y are Banach spaces and A is the generator of a C 0semigroup (T (t)) t≥0 with growth bound ω(A).The domain of A equipped with the graph norm is denoted by D(A).The spectrum of A is σ(A) and its resolvent is R(z, A) for z in the resolvent set ρ(A).The set of all bounded operators from D(A) to Y is given by L(D(A), Y ).As usually, I = I X denotes the identity operator on the space X.
Definition 2.2.Let Φ be a Young function.Then, the function Φ defined by is called the to Φ complementary Young function and Φ and Φ are called complementary to each other.
Indeed, Φ defines a Young function and Φ(t) = sup s≥0 (st − Φ(s)).For complementary Young functions Φ and Φ it holds that (see [1]) Let (Ω, F , µ) be a measure space and Y be a Banach space.Whenever Ω is an interval in R, (F , µ) refers to the Borel σ-algebra and the Lebesgue measure.
where we identify functions which coincide almost everywhere.Further, we define the so-called Luxemburg norm for f ∈ L Φ (Ω, Y ) by We define a subspace of L Φ (Ω; Y ) by We may write L Φ and E Φ if Ω and Y are clear from the context.On L Φ (Ω; Y ) we can define the Orlicz norm by where the second equality can be verified by considering dµ ≤ 1, the Orlicz norm can be interpreted as dual norm to the Luxemburg norm of the complementary Young functions.Both norms are equivalent, as for f ∈ L Φ (Ω; Y ) it holds that Indeed, the inequality is well-known for the scalar valued case, see e.g.[20].It is less obvious for vector valued Orlicz spaces as the fact that f LΦ(Ω;Y ) = f Y LΦ(Ω;R) is non-trivial.Both proofs can be found in the Appendix.Finally, L Φ (Ω; Y ) and E Φ (Ω; Y ) equipped with these norms are complete.
An important property of Young functions is the so-called ∆ 2 -condition.
We say that Φ satisfies the ∆ 2 -condition globally The Young function Φ(t) = t p with 1 < p < ∞ satisfies the ∆ 2 -condition globally.It leads to the Orlicz space L Φ = E Φ = L p and the Luxemburg norm is just the standard norm on L p .Further, the to Φ complementary Young function is (up to constants) given by Φ(s) = s p ′ with 1 p + 1 p ′ = 1 and hence L Φ = E Φ = L p ′ .Thus, Orlicz spaces and the concept of complementary Young functions generalize L p spaces and Hölder conjugates in a natural way.

Characterizing Orlicz admissibility by semigroups
We recall the definition of admissibility already indicated in the introduction.Definition 3.1.An operator C ∈ L(D(A), Y ) is called L Φ -admissible for A (or equivalently for (T (t)) t≥0 ) if for some (and hence for all) τ > 0 there exists a minimal constant c τ > 0 such that If there is no ambiguity, we will simply say that "C is (infinite-time) L Φadmissible" without referring to the semigroup.Analogously, we define (infinitetime) E Φ -admissibility.
It is not hard to show that L Φ -admissibility is invariant under scaling of the semigroup, i.e.C is L Φ -admissible for (T (t)) t≥0 if and only if C is L Φ -admissible for the rescaled semigroup (e −εt T (t)) t≥0 for any ε ∈ R. Also note that if the semigroup is exponentially stable, then finite-time and infinite-time L Φ -admissibility are equivalent.Moreover, C is infinite-time L Φ -admissible if and only if (15) holds for τ = ∞ and a constant c ∞ < ∞.The same holds for E Φ -admissibility.
The main result of this section is a generalization of a result by Callier-Grabowski [6], see also Engel [3] for a shorter proof, from L p to E Φ .
and only if for some (and hence for all) τ > 0 (τ = ∞ in the infinite-time case) the block operator matrix with domain The proof of Theorem 3.2 follows Engel's approach, relying on the following consequence from [4], and the fact that (S(t)) t≥0 on F and for some (and hence for all) t 0 > 0 it holds that sup t∈[0,t0] R(t) L(X,F ) < ∞, where R(t) is the bounded extension of the operator If one of the equivalent conditions is satisfied, the semigroup generated by A is given by .

On the Weiss conjecture for Orlicz spaces
We start by introducing some notation and assumptions in order to generalize Haak's result on the p-Weiss conjecture as stated in the introduction, see also [7], to Orlicz spaces L Φ .
We recall that A is the generator of a bounded analytic semigroup if −A is densely defined and sectorial of type ω for some ω ∈ [0, π 2 ).The latter means that σ(−A) ⊂ S ω and for every δ ∈ (ω, π) there is a constant For more about sectorial operators, the holomorphic functional calculus and the connection to bounded analytic semigroups we refer to [10] and [5].Whenever we are dealing with the generator A of a bounded analytic semigroup ω A is referred to as the angle of sectoriality of −A, that is the infimum of all ω such that −A is sectorial of type ω.Besides the holomorphic functional calculus, the functional calculus for multiplication operators (see [11,Ch. 2]) will be of interest.
The assumption of having a bounded analytic semigroup is necessary for Haak's result in the sense that if (−A) 1 /p is L p -admissible and the semigroup is bounded then it is already bounded analytic, [12,Prop. 2.7].
When passing from L p to L Φ we have to substitute both the p-Weiss condition (4) and the operator (−A) 1 /p in terms of the Young function Φ or its complementary Young function Φ.While the former is rather straight-forward, the latter is more delicate.First we generalize the p-Weiss condition (4).
for some α > 0, where Φ is the complementary Young function of Φ.We say that C satisfies the infinite-time Φ-Weiss condition if (18) holds for α = 0.
. It is obvious that the definitions of the Φ-Weiss condition and the p-Weiss condition (4) are consistent in the sense that they are the same if we consider Φ(t) = t p for 1 < p < ∞.
Similar to L p spaces it is easy to prove that (infinite-time) L Φ -admissibility of Proof.If C is infinite-time L Φ -admissible.Using the generalized Hölder inequality (8) we get for all x ∈ D(A) that holds, where c ∞ denotes the admissibility constant from Definition 3.1.If C is L Φadmissible for (T (t)) t≥0 , then C is infinite-time L Φ -admissible for the exponentially stable semigroup generated by A − α, where α > ω(A) and α > 0. Hence, the proof can be deduced from the infinite-time case.
If L Φ = L p Haak's result tells us that the converse of Lemma 4.3 holds if and only if Φ −1 (−A) = (−A) 1 /p is (infinite-time) L p -admissible, hence formally Φ −1 (−A) seems to be a suitable operator to characterize general L Φ -admissibility.However, we have to make sure that this operator is indeed a well-defined in L(D(A), X).Therefore, we make the following assumption on Φ.
Assumption 1. Assume that Φ −1 extends to a holomorphic function on some sector S δ for δ ∈ (ω A , π 2 ) and that there exist constants m 0 , m 1 > 0 such that We continue with a sequence of technical statements on the operator Φ −1 (−A) and the Φ-Weiss condition.Our approach is based on the ideas from [12], which seem to be slightly more elementary than the more natural proof of Haak using square function estimates.It seems to be a non-trivial challenge to generalize such square function estimates to the Orlicz space setting.
Recall that a multiplication operator is an operator M a : L p (Ω) → L p (Ω) for some semi-finite measure space (Ω, F , µ), 1 ≤ p ≤ ∞ and a : Ω → C measurable, given by Proof.For technical details on the functional calculus for multiplication operators and holomorphic functional calculus we will use in this proof, we refer to [11, Ch. 2] and [10, Ch. 2].Let It suffices to prove that f (−A) is bounded, where f (−A) is defined via the measurable functional calculus if we consider (i) and via the holomorphic functional calculus if we consider (ii).Indeed, we obtain from the properties of the functional calculi that in the sense of inclusion of the respective graphs of the operators.If f (−A) is bounded the operator on the left-hand side lies in L(D(A), X) and so does Φ −1 (−A).We distinguish between the two possible assumptions: (i) Since f is a bounded function on [0, ∞) we derive from the functional calculus for multiplication operators that f (−A) is bounded.(ii) To prove that f (−A) is a bounded operator, it suffices to prove that there exist c, α > 0 such that By Assumption 1, Φ −1 is holomorphic on some sector S δ and Since Φ ∈ P, we infer by ( 13) that, for |z| ≤ 1, and, by ( 14) and ( 5), that, for |z| ≥ 1, Recall Example 2.7 of Young functions of class P. While Example (iii) is only useful when A is a multiplication operator, Examples (ii) and (iv) yield Young functions Φ which satisfy Assumption 1. Example (i) tells us how to construct further examples of class P, e.g.ρ(t) = t r + log(t), r ∈ [0, 1], yields Φ ∈ P via (9) for any choice of 1 < p < q < ∞.However, in general it is not clear whether this construction leads to functions satisfying Assumption 1 again.Lemma 4.5.Suppose that A generates a bounded analytic semigroup.If either • A is a multiplication operator with σ(−A) ⊆ [0, ∞) or • Assumption 1 holds and Φ ∈ P, then we have that Proof.Let t > 0 and f (s Thus, there exists c ∈ [e −1 , 1] with sup s≥0 f (s) = c Φ −1 ( 1 t ) and the assertion follows.
Let Assumption 1 hold and let Φ be in P. Let ω, δ, m 1 be as in Assumption 1, choose δ ′ ∈ (ω, δ) and take Γ = ∂S δ ′ orientated positively.Then, where we used (13) in the last step.Since the last integral converges, the proof is complete.
Remark 4.6.We want to point out that Φ ∈ P is only needed to guarantee Φ −1 (−A) ∈ L(D(A), X) and to deal with the singularity of the integrand at 0. If we consider the integral over (ε, ∞) with ε ∈ (0, 1] we derive the estimate is decreasing and Φ −1 is increasing.Lemma 4.7.Suppose that A generates a bounded analytic semigroup.If either • A is a multiplication operator with σ(−A) ⊆ [0, ∞) or • Assumption 1 holds and Φ ∈ P and if Φ −1 (−A) is L Φ -admissible, then for every τ > 0 there exists c τ > 0 such that holds for all x ∈ D(A).
for s > 0 and f (0) = 0. Similar to Lemma 4.5 we show that tΦ −1 ( 1 t )f (−A) is uniformly bounded in t > 0. First, suppose that A is a multiplication operator.Note that f attains its maximum in [0, 2  t ], since , where we used the monotonicity of Φ −1 in the last inequality.This shows that ) and hence the claimed uniform boundedness follows.Second, consider Assumption 1 and Φ ∈ P. Let δ and m 0 be given as in Assumption 1. Choose δ ′ ∈ (ω A , δ) and let Γ = S δ ′ be orientated positively.Then, by (14).Since the last integral converges, tΦ −1 Next, we use the following identity for x ∈ D(A), x.While the first part is uniformly bounded, we can apply the admissibility estimate from Definition 3.1 to the second part to obtain the desired estimate.Note that we can decompose the operator in the above way by the properties of the functional calculus.Indeed, f (−A) is bounded and ran T ( t 2 ) ⊆ D(A) ⊆ D(Φ −1 (−A)) for all t > 0, where the first inclusion is a known fact for bounded analytic semigroups.
for some β ≥ 0, then for every τ > 0 there exist constants c τ , K τ > 0 such that holds for all x ∈ D(A).
)) is uniformly bounded by the assumptions it suffices to estimate tΦ −1 A straight-forward estimate of the Orlicz norm completes the proof.
We briefly introduce the weak Orlicz space L Φ,∞ = L Φ,∞ (0, ∞; Y ) which consists of all measurable functions f : [0, ∞) → Y (modulo functions that are zero almost everywhere) such that where f * denotes the decreasing rearrangement of f , > s} for any function g on [0, ∞).As usually, we just write L Φ,∞ (0, ∞) if Y = C.The reader is referred to [22,23] for more on weak Orlicz spaces and Orlicz-Lorentz spaces.
Proof of Theorem 4.9.To prove (i) ⇒ (ii) let δ < π 2 be larger than ω, the type of sectoriality of A and choose M > 0 such that λR(λ, A) ≤ M on S δ .We assume Γ = ∂S ′ δ to be orientated positively for δ ′ ∈ (δ, ω).For z ∈ Γ the resolvent equation yields Then, for any t > 0 and x ∈ X, we estimate where M ′ > 0 is a suitable constant according to our assumptions.For z ∈ Γ we can write Re z = |z| cos(δ ′ ) and hence, where we applied ( 14) and (5).Since the integral is finite, we are done.Next, we prove (ii) ⇒ (iii).Let M be given as in (ii).For x ∈ X we have that λ This shows that C is infinite-time L Φ,∞ -admissible.
As a direct consequence of a rescaling argument for the semigroup we can formulate the finite-time version of Theorem 4.9.
Corollary 4.10.If A generates a bounded analytic semigroup and if Φ ∈ P, then the following statements are equivalent for C ∈ L(D(A), Y ) (i) The Φ-Weiss condition (18) holds for some α > ω(A), In (i) and (ii) the parameters α and β can be chosen the same if they are nonnegative.
The approach to the proof of d's result from [12] uses the boundedness of the integral operator L, defined for τ > 0 by on L p (0, τ ; Y ) with operator norm bounded by p, see [12, Prop.2.2].As a direct consequence of the interpolation result from [18, Thm.5.1] we have the following lemma.
Lemma 4.11.If Φ ∈ P and L is given by (22) for some τ > 0 then L is a bounded operator on L Φ (0, τ ; Y ) with operator norm independent of τ > 0.
We put everything together to get our main theorem: Theorem 4.12.Suppose that A generates a bounded analytic semigroup.If Φ ∈ P and either A is a multiplication operator with σ(−A) ⊆ [0, ∞) or Assumption 1 holds.Then the following are equivalent Proof.Since A generates a bounded semigroup we have that ω(A) ≤ 0. First, assume (ii).From Lemma 4.5 and Theorem 4.9 we deduce the infinite-time Φ-Weiss condition (and hence the finite-time Φ-Weiss condition) holds for for some α > ω(A).Corollary 4.10 implies for β > max(α, 0) that where L is the integral operator given by (22), which is bounded on L Φ (0, τ ; Y ) by Lemma 4.11 since Φ ∈ P. We obtain that C(e −βt T (t))x LΦ(0,τ ;Y ) ≤ C(e −βτ T (τ ))x LΦ(0,τ ;Y ) + Lf LΦ(0,τ ;Y ) where c τ and K τ are the constants from Corollary 4.8 and L denotes the operator norm of L on L Φ (0, τ ; Y ).This shows that C is L Φ -admissible.The infinitetime case is even simpler.Assume that the infinite-time Φ-Weiss condition holds.Theorem 4.9 implies that Since L and c τ are uniformly bounded in τ > 0 (c.f.Corollary 4.8) we obtain that C is infinite-time L Φ -admissible.
On X = ℓ r , r ∈ [1, ∞), there is a sufficient condition on Φ for infinite-time L Φ -admissibility of Φ −1 (−A) when dealing with a multiplication operator A given by Ae where (e n ) n∈N is the standard basis on ℓ r and (λ n ) n is assumed to be a sequence of non-positive numbers, i.e. λ n ≤ 0 for all n ∈ N. The domain of A is given by and it is well-known that A is the generator of a bounded analytic semigroup (T (t)) t≥0 given by T (t)e n = e λnt e n , n ∈ N. Clearly, for any Young function Φ the functional calculus for multiplication operators yields that Φ −1 (−A) is given by Proposition 4.13.Consider the operator A on ℓ r given by (23).
Proof.Similar to the L p case we obtain for holds, where we applied the generalized Minkowski inequality (Proposition 6.8) and Lemma 6.2.This proves that Φ −1 (−A) is infinite-time L Φ -admissible.
Remark 4.14.We would like to point out that the theory developed in this section is also applicable to selfadjoint operators A on Hilbert spaces.Indeed, by the spectral theorem (see [10, Thm.D.5.1])A is unitary equivalent to a multiplication operator and admissibility of C for A is preserved under unitary transformations of C and A.

Duality of control and observation operators
The previous results on Orlicz admissibility for observation operators can easily be transfered to control operators via duality.We extend known duality results for L p , see e.g.[27], to Orlicz spaces.
Let X, U, Y be Banach spaces such that A is the generator of a C 0 -semigroup (T (t)) t≥0 on X and the dual semigroup (T ′ (t)) t≥0 is also strongly continuous.This is in particular the case if X is reflexive.By X 1 we denote D(A) equipped with the graph norm of A and X −1 is the completion of X with respect to the norm defined by x X−1 = (β − A) −1 x X , x ∈ X for some β ∈ ρ(A).Different choices of β ∈ ρ(A) leads to equivalent norms on X −1 .The C 0 -semigroup (T (t)) t≥0 has a unique extension to a C 0 semigroup (T −1 (t)) t≥0 on X −1 whose generator is an extension of A. The same construction can be done for A ′ which leads to spaces X d 1 and X d −1 .We have continuous and dense embeddings X 1 ֒→ X ֒→ X −1 and X d 1 ֒→ X ′ ֒→ X d −1 , and dual pairings y which are nothing else as the standard dual pairing on X ′ and X if y 1 ∈ X ′ and x 2 ∈ X, see e.g.[25].The spaces X −1 and X d −1 are the dual spaces of X d 1 and X 1 with respect to the pivot spaces X and X ′ respectively ( [27,Ch. 6]).This concept gives rise to dual operators for B ∈ L(U, X −1 ) and ).Thus a control operator B ∈ L(U, X −1 ) of the control system ẋ(t) = Ax(t) + Bu(t), t ≥ 0, x(0) = x 0 , can be viewed as a observation operator of the dual observation system in the sense of (1) in X ′ with A ′ instead of A. In the same manner C ′ can be seen as a control operator of a system in X ′ .Definition 5.1.The control operator B ∈ L(U, X −1 ) is called L Φ -admissible (for A or (T (t)) t≥0 ) if for some (and hence for all) t > 0 there exists a minimal constant b τ > 0 such that We give the following duality result, which extends Weiss' result for L p [27].
1.The observation operator ) is a (infinite-time) L Φ -admissible control operator.Moreover, the admissibility constants coincide.

If the control operator
) is a (infinite-time) L Φ -admissible observation operator.Moreover, denoting the admissibility constants of B and B ′ by b τ and c τ , we have that c τ ≤ b τ .Equivalence holds if U is reflexive, in this case the admissibility constants coincide.
Proof.By Proposition 6.3 the proof is analogous to the one of [27,Thm. 6.3].

Appendix: Further results on Orlicz spaces
We give some technical results on Orlicz spaces, some of which we could not locate in the literature, and hence may be of interest in their own right.This includes Proposition 6.4, Proposition 6.5, Proposition 6.6 and Proposition 6.8.The functions Φ and Φ will always be complementary Young functions and (Ω, F , µ) will denote a measure space.
We start off with a technical result on the class P, c.f. Section 2 and the Orlicz norm of parameter depending exponential functions which is of interest when working with the Laplace transform representation of the resolvent of an operator.Lemma 6.1.Let f : (0, ∞) → (0, ∞) be given by for t > 0, where ρ : (0, ∞) → (0, ∞) is a continuous concave function such that ρ(st) ≤ max(1, s)ρ(t) for all s, t > 0. Then f is strictly increasing and hence invertible on (0, ∞).Its inverse f −1 is a Young function.
Thus, to derive |ρ(z)| ≤ mρ(|z|) for some positive constant m independent of z we can show that We continue with some more general results on Orlicz spaces and the standard norms on them.
Thus, since ε and h was arbitrary the assertion for f follows.The proof for g follows similarly.
By the definition of the Orlicz norm we obtain f LΦ(Ω;Y ) = f (•) Y LΦ(Ω) and thus, we deduce from the scalar valued case Y = R, see e.g.[20], that , where we applied the previous derived estimate for r = 1 and the Young function Ψ.

Definition 2 . 3 .
Let Φ be a Young function.The Orlicz space L Φ (Ω; Y ) is given by L Φ (Ω; Y ) := f : Ω → Y f is Bochner measurable and for some k > 0 : generates the right shift semigroup on E Φ (0, τ ; Y ), see Proposition 6.5.Lemma 3.3.If X and F are Banach spaces, A : D(A) ⊆ X → X and D : D(D) ⊆ F → F are closed and densely defined operators such that (ω, ∞) ⊆ ρ(A) ∩ ρ(D) for some ω ∈ R and if L ∈ L(D(A), F ), then the following are equivalent (a) The block operator matrix

Definition 4 . 1 .
An operator C ∈ L(D(A), Y ) is said to satisfy the Φ-Weiss condition, for a Young function Φ, if

Corollary 4 . 8 .
Suppose that A generates a bounded analytic semigroup.If either • A is a multiplication operator with σ(−A) ⊆ [0, ∞) or • Assumption 1 holds and Φ ∈ P and if Φ