THE MAXIMAL SUBSPACE FOR GENERATION OF ( a ; k ) − REGULARIZED FAMILIES

A BSTRACT . Let A be a linear operator in a Banach space X . We deﬁne a subspace of X and a norm such that the part of A in such subspace generates an ( a ; k ) − regularized resolvent family. This space is maximal-unique in a suitable sense and non-trivial, under certain conditions on the kernels a and k .


INTRODUCTION
Inspired in the well known Hille-Yosida theorem, Kantorovitz defined in 1988 a linear subspace and a norm such that the restriction of A to this subspace generates a strongly continuous semigroup of contractions (see [12]).This so-called Hille-Yosida space is maximal unique in a suitable sense.The same problem has been considered in the context of strongly continuous operator families of contractions by Cioranescu in [5].In this case, the generation theorem of Sova and Fattorini was fundamental for her work.Later, Lizama in [16] used the generation theorem for resolvent families due to Da Prato and Ianelli (see [6]) as basis to generalize the results of Kantorovitz and Cioranescu to the context of resolvent families of bounded and linear operators.In this paper also some applications to Volterra equations were given.
It is remarkable that resolvent families do not include α−times integrated semigroups, α−times integrated cosine functions, K−convoluted semigroups, K−convoluted cosine families and integrated Volterra equations, among others.For an historical account of these classes of operators, see [1, p.234].Actually, these type of families are (a, k)−regularized.The concept of (a, k)− regularized resolvent families was introduced in [17].The systematic treatment based on techniques of Laplace transforms was developed in several papers (see e.g.[14], [19], [23], [21], [22], [24] and [25].)The theory of (a, k)-regularized families has been developed in many directions and we refer to the recent monograph of Kostic [15] for further information.In this context, the problem to find maximal subspaces for generation of (a, k)−regularized families remained open in case k(t) ̸ = 1.In this paper, we are able to close this gap generalizing, in particular, the papers [5], [12] and [17].
In this work, we will use the generation theorem for (a, k)− regularized resolvent family (see [17]) to show that there exists a linear subspace Z a,k in X and a norm | • | a,k majorizing the given norm, such that (Z a,k , | • | a,k ) is a Banach space, and the part of A in Z a,k generates a (a, k)− regularized resolvent family of contractions in Z a,k .Moreover, the space (Z a,k , | • | a,k ) is a maximal-unique in a sense to be defined below.Concerning the non-triviality of Z a,k we prove that it contains the eigenvectors corresponding to non-positive eigenvalues of A. We close this paper with illustrative examples concerning the cases a(t) = t α−1 and k(t) = t β in some region α > 0 and β > 0.
This paper is organized as follows.In the first section, we recall the definition as well as basic results about (a, k)-regularized families.
In section 2, we show the existence of the maximal subspace such that the part of A in this subspace generates an (a, k)-regularized family.We prove that such subspace is a Banach space with the norm defined below.The maximality is also proved and we show how this is used for obtain a relation with the Hille-Yosida space corresponding to the semigroup case.
In section 3, we present some applications of the theory developed in preceding section.Here we show the particular cases of generation corresponding to resolvent families, cosine operator families, semigroups, α−times semigroups and α−times cosine operator families.After that, we give concrete conditions on a given operator A to obtain the non-triviality of the maximal spaces and hence the well posedness on these spaces, for the abstract Cauchy problems of first and second order.

PRELIMINARIES
In this section we recall some useful results in the literature about (a, k)− regularized resolvent families.Let us fix some notations.From now on, we take X to be a Banach space with the norm ∥ • ∥.We denote by B(X) the Banach algebra of all bounded linear operators on X endowed with the operator norm, which again is denoted by ∥ • ∥.The identity operator on X is denoted by I ∈ B(X), and R + denotes the interval [0, ∞).For a closed operator A, we denote by σ (A), σ p (A), ρ(A) the spectrum, the point spectrum and resolvent of A respectively.Definition 2.1.Let k ∈ C(R + ), k ̸ = 0 and a ∈ L 1 loc (R + ) be given.Assume that A is a linear operator with domain D(A).A strongly continuous family {R(t)} t≥0 ⊂ B(X) is called (a, k)−regularized family on X having A as a generator if the following hold: xds for all t ≥ 0 and x ∈ D(A).In the case where k(t) ≡ 1, this definition corresponds to the resolvent family for the Volterra equation of convolution type in [17].Moreover, if in addition, a(t) ≡ 1 then this family is a C 0 −semigroup on X or if a(t) ≡ t is a cosine family on X.
We note that the study of (a, k)−regularized families is associated to a wide class of linear evolution equation, including e.g.fractional abstract differential equations (see [20]).Definition 2.2.We say that (R(t)) t≥0 is of type (M, ω) if there exist constants M ≥ 0 and ω ∈ R such that We will require the following theorem on generation of (a, k)−regularized families (see [17]).
Theorem 2.3.Let A be a closed and densely defined operator on a Banach space X.Then {R(t)} t≥0 is an (a, k)−regularized family of type (M, ω) if and only if the following hold: In the case where k(t) ≡ 1, the Theorem 2.3 is well known.In fact, if a(t) ≡ 1 then it is just the Hille-Yosida Theorem; If a(t) ≡ t, then it is the generation theorem due essentially to Da Prato and Iannelli in [6].In the case where k(t) = t n n! and a(t) ≡ 1, is the generation theorem for n−times integrated semigroups [13];k(t) = t n n! and a(t) is arbitrary, it corresponds to generation theorem for integrated solutions of Volterra equations due to Arendt and Kellerman [2].
In order to give applications to our results we recall the following concepts of fractional calculus.The Mittag-Leffler function (see e.g.[7,9] and [8]) is defined as follows: where H a is a Hankel path, i.e. a contour which starts and ends at −∞ and encircles the disc |µ| ≤ |z| 1/α counterclockwise.The function E α,β is an entire function which provides a generalization of several usual functions.For a recent review, we refer to the monograph [11].

THE MAXIMAL SUBSPACE
In this section, X is a Banach space with norm ∥ • ∥.Let A be a linear operator and a ∈ L and 1 â(λ ) ∈ ρ(A) for all λ > 0. Observe that we are implicitly assuming that the inequality (2.1) holds with ω = 0.
) −1 for λ > 0, and define where for l = 0 the product is defined as First, we show that x ∈ Y a,k .Indeed, let λ j > 0, n j , l ∈ N 0 be fixed.Then Second, we prove that {x i } converges to x ∈ Y a,k .Let ε > 0. There exists where M > 0 is the constant given in (2.1).Hence for every λ j > 0, n j , l ∈ N 0 we have by inequality (2.1): Taking supremum over all λ j > 0, n j , l ∈ N 0 we obtain Therefore the sequence

This operator is sometimes called the part of
therefore H a,k (λ )y ∈ Y a,k .On the other hand, from identities above ), and we conclude that Now if y ∈ D(A Y a,k ) then, in particular, y ∈ D(A), and therefore, This proves the second assertion.In particular, ρ(A Y a,k ) ̸ = / 0 and hence Finally, let y ∈ Y a,k , λ > 0, n ∈ N 0 be fixed.We have where λ j > 0 are arbitrary for 1 ≤ j ≤ l, λ l+1 := λ and n l+1 := n.This proves the third part of the lemma.
λ n+1 for each λ > 0 and n ∈ N 0 where H a,k (λ Proof.We observe that H a,k (λ ) = H a,y (λ )| Z a,k .Then the result is a direct consequence of [1,Lemma 3.3.12].
As a consequence, we obtain the main result of this section on the existence of (a, k)−regularized families.
Theorem 3.5.Let A a,k be a linear operator defined in a Banach space X and a ∈ L 1 loc (R + ) and k ∈ C(R + ), k ̸ = 0. Assume that 1 â(λ ) ∈ ρ(A) for all λ > 0. Then there exist a linear subspace Z a,k and a norm ) is a Banach space and A a,k generates an (a, k)−regularized family of contractions in Z a,k .
Proof.According with our hypothesis, we can apply the generation theorem for (a, k)−regularized family (see [17,Th. 3.4]) and the result of Lemma 3.4.
On concerning to the non-triviality of Z a,k , we will prove that it contains the eigenvectors corresponding to non-positive eigenvalues of A.
Let µ ∈ C be fixed.Let r(t, µ) be the unique solution to the scalar equation Thus,provided the kernels a(t) and k(t) are Laplace transformable, we have We define C a,k := {µ ∈ σ (A) : the map t → r(t, µ), t ≥ 0 is bounded} .
Proposition 3.6.Let x be an eigenvector of A corresponding to the eigenvalue α ∈ C a,k .Then x ∈ Z a,k .
Proof.Let x be a eigenvector of A corresponding to the eigenvalue α such that the map t → r(t, α) is bounded.
Let λ > 0 and n ∈ N be fixed.Then This implies that |x| a,k ≤ sup t≥0 ∥r(t, α)∥∥x∥ and consequently, x ∈ Z a,k .
The following result shows us that the spaces Z a,k are maximal-unique in a certain sense.
Theorem 3.7.Under the same hypothesis of Theorem 3.5, if Finally, this implies that D(B a,k ) ⊂ D(A a,k ) and B a,k ⊂ A a,k .
The next result treats about the "maximal property".In order to obtain the analogous result to the resolvent families case we need more information about the function k(t).Theorem 3.8.Let A be a linear operator defined in a Banach space X and k ∈ C 1 (R + ), k ̸ = 0 with absolutely convergent Laplace transform for λ > 0. Assume that 1 â(λ ) ∈ ρ(A) for all λ > 0. Suppose that 1 ≤ k(0).Then A a,k generates a strongly continuous semigroup of contractions on Z a,k .
Remark 3.9.From the maximal uniqueness of Z Taking k(t) ≡ 1 or k(t) ≡ t we obtain from the preceding corollary the following.

Corollary 4.2. [12]
Let A be a linear operator on X such that (0, ∞) ⊂ ρ(A).Then there exist a linear subspace Z ) is a Banach space and the restriction A 1 of A to Z 1 is the infinitesimal generator of a C 0 −semigroup of contractions on Z 1 .

Corollary 4.3. [5]
Let A be a linear operator on X such that (0, ∞) ⊂ ρ(A).Let A t be the operator in Z t defined as above.Then A t is the infinitesimal generator of a strongly continuous cosine family of contractions on Z t .Remark 4.4.Applying the Theorem 3.5 with k(t) ≡ t β Γ(β +1) we obtain corresponding results for α-times integrated semigroups and β -times integrated cosine families taking a(t) ≡ 1 and a(t) ≡ t respectively.Suppose that σ p (A) ̸ = / 0 and µ ∈ σ p (A).In the following examples we search conditions under which the function t → r(t, µ) is bounded.This ensure the non-triviality of the subspace Z a,k .It can be verified the following formulas (see [10,Formula 2.632]) ] .
Note that arg(iωt) = π 2 and arg(iωt) = − π 2 ; so we can apply (4.4).Then, for t sufficiently large, we obtain that The boundedness follows from the fact that β < 1.Therefore, for t sufficiently large where K depends of ω and β .Since (0, ∞) ⊂ ρ(A) we have that λ 2 ∈ ρ(A) for λ > 0. It follows from Theorem 3.5 that A is the infinitesimal generator of β −times integrated cosine function (C β (t)) t≥0 of contractions on Z 2,β for 0 < β ≤ 1.As in the above example, it now means that the initial value problem (4.5) is well posed in the sense that there exists a strongly continuous family of linear operators (C β (t)) t≥0 on a nontrivial subspace Z 2,β of X such that for all initial values u 0 ∈ Z 2,β there exists a unique classical solution of (4.5).
A in Y a,k .We denote Z a,k := D(A Y a,k ), where the closure is taken in the norm | • | a,k .Lemma 3.3.With the preceding definitions and hypothesis, we have: (a) A Y a,k is a closed linear operator on Y a,k .− A Y a,k is invertible on Y a,k for each λ > 0. (c) ∥H (n) a,y (λ )∥ B(Y a,k ) ≤ n! λ n+1 for each λ > 0 and n ∈ N 0 where H a,y (λ ) := k(λ )