The Maximal Subspace for Generation of (𝑎,𝑘)-Regularized Families

Abstract

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 nontrivial, under certain conditions on the kernels 𝑎 and 𝑘.

1. Introduction

Inspired by 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 [1]). 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 [2]. In this case, the generation theorem of Sova and Fattorini was fundamental for her work. Later, Lizama in [3] used the generation theorem for resolvent families due to Da Prato and Iannelli (see [4]) 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. For a historical account of these classes of operators, see ([5], page 234). Actually, these types of families are (𝑎,𝑘)-regularized. The concept of (𝑎,𝑘)-regularized resolvent families was introduced in [6]. The systematic treatment based on techniques of Laplace transforms was developed in several papers (see, e.g., [7–13]). The theory of (𝑎,𝑘)-regularized families has been developed in many directions and we refer to the recent monograph of Kostić [14] 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, [1, 2, 6].

In this work, we will use the generation theorem for (𝑎,𝑘)-regularized resolvent family (see [6]) 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 to 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 the 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.

2. 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∈𝐿¹_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:(a)𝑅(0)=𝑘(0)𝐼; (b)𝑅(𝑡)𝑥∈𝐷(𝐴) and 𝑅(𝑡)𝐴𝑥=𝐴𝑅(𝑡)𝑥 for all 𝑥∈𝐷(𝐴) and 𝑡≥0; (c)𝑅(𝑡)𝑥=𝑘(𝑡)𝑥+∫^𝑡₀𝑎(𝑡−𝑠)𝐴𝑅(𝑠)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 [6]. Moreover, if, in addition, 𝑎(𝑡)≡1 then this family is a 𝐶₀-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, for example, fractional abstract differential equations (see [15]).

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 ‖𝑅(𝑡)‖≤𝑀𝑒^𝜔𝑡 (2.1) for all 𝑡≥0.

We will require the following theorem on generation of (𝑎,𝑘)-regularized families (see [6]).

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: (a)𝑎(𝜆)≠0 and 1/𝑎(𝜆)∈𝜌(𝐴) for all 𝜆>𝜔;(b)𝐻(𝜆)∶=𝑘(𝜆)(𝐼−𝑎(𝜆)𝐴)⁻¹ satisfies the estimates 𝐻^(𝑛)(𝜆)≤𝑀𝑛!(𝜆−𝜔)^𝑛+1,𝜆>𝜔,𝑛∈ℕ₀. (2.2)

In the case where 𝑘(𝑡)≡1, 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 [4]. In the case where 𝑘(𝑡)=𝑡^𝑛/𝑛! and 𝑎(𝑡)≡1, it is the generation theorem for 𝑛-times integrated semigroups [16]; if 𝑘(𝑡)=𝑡^𝑛/𝑛! and 𝑎(𝑡) is arbitrary, it corresponds to the generation theorem for integrated solutions of Volterra equations due to Arendt and Kellerman [17].

In order to give applications to our results we recall the following concepts of fractional calculus. The Mittag-Leffler function (see, e.g., [18–20]) is defined as follows: 𝐸_𝛼,𝛽(𝑧)∶=^∞_𝑛=0𝑧^𝑛Γ=1𝛼𝑛+𝛽2𝜋𝑖_𝐻𝑎𝑒^𝜇𝜇^𝛼−𝛽𝜇^𝛼−𝑧𝑑𝜇,𝛼,𝛽>0,𝑧∈ℂ, (2.3) where 𝐻_𝑎 is a Hankel path, that is, a contour which starts and ends at −∞ and encircles the disc |𝜇|≤|𝑧|^1/𝛼 counter clockwise. 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 [21].

An interesting property related with the Laplace transform of the Mittag-Leffler function is the following (cf. [18], (A.27) page 267): ℒ𝑡^𝛽−1𝐸_𝛼,𝛽−𝜌^𝛼𝑡^𝛼(𝜆)=𝜆^𝛼−𝛽𝜆^𝛼+𝜌^𝛼,Re𝜆>𝜌^1/𝛼;𝛼>0,𝛽>0,𝜌∈ℝ. (2.4)

Remark 2.4 (see [22]). If 0<𝛼<2 and 𝛽>0, then 𝐸_𝛼,𝛽(𝑧)=1𝛼𝑧^{(1−𝛽)/𝛼}exp𝑧^1/𝛼+𝜀_𝛼,𝛽(𝑧),(𝑧≤1arg)2𝐸𝛼𝜋;_𝛼,𝛽(𝑧)=𝜀_𝛼,𝛽(𝑧),(<1arg−𝑧)1−2𝛼𝜋, (2.5) where 𝜀_𝛼,𝛽(𝑧)∶=−^𝑁−1_𝑛=1𝑧^−𝑛Γ|𝑧|𝛽−𝛼𝑛+𝒪^−𝑁, (2.6) as 𝑧→∞ with 2≤𝑁∈ℕ. This implies that for each 𝜋(𝛼/2)<𝜔<min{𝜋,𝜋𝛼}, there is a constant 𝐶∶=𝐶(𝜔)>0 such that 𝐸_𝛼,𝛽(𝑧≤𝐶)|𝑧|(𝑧1+,𝜔≤arg)≤𝜋. (2.7)

3. The Maximal Subspace

In this section, 𝑋 is a Banach space with norm ‖⋅‖. Let 𝐴 be a linear operator and 𝑎∈𝐿¹_loc(ℝ₊) and 𝑘∈𝐶(ℝ₊), 𝑘≠0. Assume that 𝑎(𝜆)=^∞₀𝑒^−𝜆𝑡𝑎(𝑡)𝑑𝑡≠0,∀𝜆>0, (3.1) and 1/𝑎(𝜆)∈𝜌(𝐴) for all 𝜆>0. Observe that we are implicitly assuming that the inequality (2.2) holds with 𝜔=0. Let 𝐻_𝑎,𝑘(𝜆)∶=(𝑘(𝜆)/𝑎(𝜆))((1/𝑎(𝜆))−𝐴)⁻¹ for 𝜆>0, and define 𝑌_𝑎,𝑘∶=𝑥∈𝑋∶|𝑥|_𝑎,𝑘∶=sup_{𝜆𝑗>0,𝑛𝑗,𝑙∈ℕ0}^𝑙_𝑗=11𝑛_𝑗!𝜆^𝑛𝑗_𝑗⁺¹𝐻^(𝑛𝑗)_𝑎,𝑘𝜆_𝑗𝑥<+∞, (3.2) where for 𝑙=0 the product is defined as 𝑥. It is clear that |⋅|_𝑎,𝑘 is a norm on 𝑌_𝑎,𝑘.

Proposition 3.1. (𝑌_𝑎,𝑘,|⋅|_𝑎,𝑘) is a Banach space.

Proof. Let {𝑥_𝑖}⊂𝑌_𝑎,𝑘 be a Cauchy sequence. We observe that ‖𝑥‖≤|𝑥|_𝑎,𝑘 if 𝑥∈𝑋. Then {𝑥_𝑖} is a Cauchy sequence on 𝑋. Let 𝑥∶=lim_𝑖→∞𝑥_𝑖.
First, we show that 𝑥∈𝑌_𝑎,𝑘. Indeed, let 𝜆_𝑗>0, 𝑛_𝑗,𝑙∈ℕ₀ be fixed. Then ^𝑙_𝑗=11𝑛_𝑗!𝜆^𝑛𝑗_𝑗⁺¹𝐻^(𝑛𝑗)_𝑎,𝑘𝜆_𝑗𝑥=lim_𝑖→∞^𝑙_𝑗=11𝑛_𝑗!𝜆^𝑛𝑗_𝑗⁺¹𝐻^(𝑛𝑗)_𝑎,𝑘𝜆_𝑗𝑥_𝑖≤limsup_𝑖→∞𝑥_{𝑖𝑎,𝑘}<+∞. (3.3)
Second, we prove that {𝑥_𝑖} converges to 𝑥∈𝑌_𝑎,𝑘. Let 𝜀>0. There exists 𝑁∶=𝑁(𝜀)∈ℕ such that |𝑥_𝑖−𝑥_𝑚|_𝑎,𝑘<𝜀/2 for 𝑖,𝑚>𝑁. Since 𝑥=lim_𝑖→∞𝑥_𝑖 in the norm of 𝑋, we also have  ‖𝑥_𝑚−𝑥‖<𝜀/2𝑀 where 𝑀>0 is the constant given in (2.2). Hence for every 𝜆_𝑗>0, 𝑛_𝑗,𝑙∈ℕ₀, we have by inequality (2.2) ^𝑙_𝑗=11𝑛_𝑗!𝜆^𝑛𝑗_𝑗⁺¹𝐻^(𝑛𝑗)_𝑎,𝑘𝜆_𝑗𝑥_𝑖−𝑥≤^𝑙_𝑗=11𝑛_𝑗!𝜆^𝑛𝑗_𝑗⁺¹𝐻^(𝑛𝑗)_𝑎,𝑘𝜆_𝑗𝑥_𝑖−𝑥_𝑚+^𝑙_𝑗=11𝑛_𝑗!𝜆^𝑛𝑗_𝑗⁺¹𝐻^(𝑛𝑗)_𝑎,𝑘𝜆_𝑗𝑥_𝑚≤−𝑥_𝑙𝑗=11𝑛_𝑗!𝜆^𝑛𝑗_𝑗⁺¹𝐻^(𝑛𝑗)_𝑎,𝑘𝜆_𝑗𝑥_𝑖−𝑥_𝑚+^𝑙_𝑗=11𝑛_𝑗!𝜆^𝑛𝑗_𝑗⁺¹𝐻^(𝑛𝑗)_𝑎,𝑘𝜆_𝑗𝑥_𝑚≤−𝑥_𝑙𝑗=11𝑛_𝑗!𝜆^𝑛𝑗_𝑗⁺¹𝐻^(𝑛𝑗)_𝑎,𝑘𝜆_𝑗𝑥_𝑖−𝑥_𝑚+𝑀𝑥_𝑚−𝑥,𝑖,𝑚>𝑁. (3.4)
Taking supremum over all 𝜆_𝑗>0, 𝑛_𝑗,𝑙∈ℕ₀, we obtain 𝑥_𝑖−𝑥_𝑎,𝑘≤𝑥_𝑖−𝑥_{𝑚𝑎,𝑘}+𝑀𝑥_𝑚<𝜀−𝑥2+𝜀2=𝜀,𝑖>𝑁. (3.5)
Therefore the sequence {𝑥_𝑖} converges to 𝑥∈𝑌_𝑎,𝑘 in the norm |⋅|_𝑎,𝑘. Consequently (𝑌_𝑎,𝑘,|⋅|_𝑎,𝑘) is a Banach space.

Definition 3.2. Let 𝐴_{𝑌𝑎,𝑘}∶𝐷(𝐴_{𝑌𝑎,𝑘})⊆𝑌_𝑎,𝑘→𝑌_𝑎,𝑘 be defined by 𝐴_{𝑌𝑎,𝑘}𝑥∶=𝐴𝑥, where 𝐷𝐴_{𝑌𝑎,𝑘}∶=𝑥∈𝐷(𝐴)∶𝑥,𝐴𝑥∈𝑌_𝑎,𝑘. (3.6)
This operator is sometimes called the part of 𝐴 in 𝑌_𝑎,𝑘.
We denote 𝑍_𝑎,𝑘∶=𝐷𝐴_{𝑌𝑎,𝑘}, (3.7) where the closure is taken in the norm |⋅|_𝑎,𝑘.

Lemma 3.3. With the preceding definitions and hypothesis, we have(a)𝐴_{𝑌𝑎,𝑘} is a closed linear operator on 𝑌_𝑎,𝑘,(b)(1/𝑎(𝜆))−𝐴_{𝑌𝑎,𝑘} is invertible on 𝑌_𝑎,𝑘 for each 𝜆>0, (c)‖𝐻^(𝑛)_𝑎,𝑦(𝜆)‖_{𝐵(𝑌𝑎,𝑘)}≤𝑛!/𝜆^𝑛+1 for each 𝜆>0 and 𝑛∈ℕ₀, where 𝐻_𝑎,𝑦(𝜆)∶=(𝑘(𝜆)/𝑎(𝜆))((1/𝑎(𝜆))−𝐴_{𝑌𝑎,𝑘})⁻¹.

Proof. Let 𝜆>0 be fixed. Since 1/𝑎(𝜆)∈𝜌(𝐴), then 𝑎(𝜆)𝑘(𝜆)1𝑎(𝜆)−𝐴𝐻_𝑎,𝑘(𝜆)𝑥=𝑥,foreach𝐻𝑥∈𝑋_𝑎,𝑘(𝜆)𝑎(𝜆)𝑘(𝜆)1𝑎(𝜆)−𝐴𝑥=𝑥,foreach𝑥∈𝐷(𝐴). (3.8)
Note that 𝐴_{𝑌𝑎,𝑘} is closed because 𝑌_𝑎,𝑘 is a Banach space and 𝐴 is closed. So the first part is done. Now, let 𝑦∈𝑌_𝑎,𝑘 be fixed. Since 𝐻_𝑎,𝑘(𝜆)∶𝑋→𝐷(𝐴), then 𝐻_𝑎,𝑘(𝜆)𝑦∈𝐷(𝐴). Moreover, 𝐻_𝑎,𝑘(𝜆)𝑦_𝑎,𝑘∶=sup^𝑙_𝑗=11𝑛_𝑗!𝜆^𝑛𝑗_𝑗⁺¹𝐻^𝑛𝑗_𝑎,𝑘𝜆_𝑗𝐻_𝑎,𝑘(𝜆)𝑦≤𝐻_𝑎,𝑘(𝜆)sup^𝑙_𝑗=11𝑛_𝑗!𝜆^𝑛𝑗_𝑗⁺¹𝐻^𝑛𝑗_𝑎,𝑘𝜆_𝑗𝑦=𝐻_𝑎,𝑘(𝜆)𝑦_𝑎,𝑘<∞, (3.9) therefore 𝐻_𝑎,𝑘(𝜆)𝑦∈𝑌_𝑎,𝑘. On the other hand, from identities above, we have 𝐴𝐻_𝑎,𝑘(𝜆)1𝑦=𝑎(𝜆)𝐻_𝑎,𝑘(𝜆)𝑘(𝜆)𝑦−𝑎(𝜆)𝑦; (3.10) thus 𝐴𝐻_𝑎,𝑘(𝜆)𝑦∈𝑌_𝑎,𝑘. Hence 𝐻_𝑎,𝑘(𝜆)𝑦∈𝐷(𝐴_{𝑌𝑎,𝑘}), and we conclude that 𝑎(𝜆)𝑘(𝜆)1𝑎(𝜆)−𝐴_{𝑌𝑎,𝑘}𝐻_𝑎,𝑘(𝜆)𝑦=𝑦,foreach𝑦∈𝑌_𝑎,𝑘. (3.11) Now if 𝑦∈𝐷(𝐴_{𝑌𝑎,𝑘}) then, in particular, 𝑦∈𝐷(𝐴), and therefore, 𝐻_𝑎,𝑘(𝜆)𝑎(𝜆)𝑘(𝜆)1𝑎(𝜆)−𝐴_{𝑌𝑎,𝑘}𝑦=𝑦. (3.12) This proves the second assertion. In particular, 𝜌(𝐴_{𝑌𝑎,𝑘})≠∅ and hence 𝐻_𝑎,𝑦(𝜆)=𝐻_𝑎,𝑘(𝜆)∣_{𝑌𝑎,𝑘},∀𝜆>0. (3.13)
Finally, let 𝑦∈𝑌_𝑎,𝑘, 𝜆>0, 𝑛∈ℕ₀ be fixed. We have1𝜆𝑛!^𝑛+1𝐻_𝑎,𝑦(𝜆)𝑦_𝑎,𝑘∶=sup^𝑙_𝑗=11𝑛_𝑗!𝜆^𝑛𝑗_𝑗⁺¹𝐻^𝑛𝑗_𝑎,𝑘𝜆_𝑗1𝜆𝑛!^𝑛+1𝐻_𝑎,𝑦(𝜆)𝑦≤sup^𝑙+1_𝑗=11𝑛_𝑗!𝜆^𝑛𝑗_𝑗⁺¹𝐻^(𝑛𝑗)_𝑎,𝑦𝜆_𝑗𝑦≤𝑦_𝑎,𝑘, (3.14) 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.

Lemma 3.4. Let 𝐴_𝑎,𝑘∶𝐷(𝐴_𝑎,𝑘)⊆𝑍_𝑎,𝑘→𝑍_𝑎,𝑘 be defined by 𝐴_𝑎,𝑘𝑥∶=𝐴_{𝑌𝑎,𝑘}𝑥, where 𝐷𝐴_𝑎,𝑘∶=𝑥∈𝐷𝐴_{𝑌𝑎,𝑘}∶𝑥,𝐴_{𝑌𝑎,𝑘}𝑥∈𝑍_𝑎,𝑘. (3.15) Then 𝐴_𝑎,𝑘 is a closed operator such that 𝐷(𝐴_𝑎,𝑘)=𝑍_𝑎,𝑘 and (a)(1/𝑎(𝜆))−𝐴_𝑎,𝑘 is invertible on 𝑍_𝑎,𝑘 for each 𝜆>0, (b)‖𝐻^(𝑛)_𝑎,𝑘(𝜆)‖_{𝐵(𝑍𝑎,𝑘)}≤𝑛!/𝜆^𝑛+1 for each 𝜆>0 and 𝑛∈ℕ₀ where 𝐻_𝑎,𝑘(𝜆)∶=(𝑘(𝜆)/𝑎(𝜆))((1/𝑎(𝜆))−𝐴_𝑎,𝑘)⁻¹.

Proof. We observe that 𝐻_𝑎,𝑘(𝜆)=𝐻_𝑎,𝑦(𝜆)∣_{𝑍𝑎,𝑘}. Then the result is a direct consequence of ([5], Lemma  3.10.2).

As a consequence, we obtain the main result of this section on the existence of (𝑎,𝑘)-regularized families.

Theorem 3.5. Let 𝐴 be a linear operator defined in a Banach space 𝑋 and 𝑎∈𝐿¹_loc(ℝ₊) and 𝑘∈𝐶(ℝ₊), 𝑘≠0. Assume that 1/𝑎(𝜆)∈𝜌(𝐴) for all 𝜆>0. Then there exist a linear subspace 𝑍_𝑎,𝑘 and a norm |⋅|_𝑎,𝑘 such that (𝑍_𝑎,𝑘,|⋅|_𝑎,𝑘) is a Banach space and 𝐴_𝑎,𝑘 generates an (𝑎,𝑘)-regularized family of contractions in 𝑍_𝑎,𝑘.

Proof. According to our hypothesis, we can apply the generation theorem for (𝑎,𝑘)-regularized family (see [6, Theorem  3.4]) and the result of Lemma 3.4.

Concerning the non-triviality of 𝑍_𝑎,𝑘, we will prove that it contains the eigenvectors corresponding to nonpositive eigenvalues of 𝐴.

Let 𝜇∈ℂ be fixed. Let 𝑟(𝑡,𝜇) be the unique solution to the scalar equation 𝑟𝑡,𝜇=𝑘(𝑡)+𝜇^𝑡₀𝑎(𝑡−𝑠)𝑟𝑠,𝜇𝑑𝑠. (3.16)

Thus, provided the kernels 𝑎(𝑡) and 𝑘(𝑡) are Laplace transformable, we have 𝐻_𝑎,𝑘(𝜆)=𝑘(𝜆)𝑎(𝜆)1𝑎(𝜆)−𝜇⁻¹=^∞₀𝑒^−𝜆𝑡𝑟𝑡,𝜇𝑑𝑡. (3.17)

We define 𝐶_𝑎,𝑘∶=𝜇∈𝜎(𝐴)∶themap𝑡→𝑟𝑡,𝜇,𝑡≥0isbounded. (3.18)

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 an 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 1𝜆𝑛!^𝑛+1𝐻^(𝑛)_𝑎,𝑘(𝜆)𝑥=1𝜆𝑛!^𝑛+1𝑑^𝑛𝑑𝜆^𝑛𝑘(𝜆)𝑎(𝜆)1𝑎(𝜆)−𝛼⁻¹𝑥=1𝜆𝑛!^𝑛+1𝑑^𝑛𝑑𝜆^𝑛∞₀𝑒^−𝜆𝑡𝑟()𝑥≤1𝑡,𝛼𝑑𝑡𝜆𝑛!^𝑛+1∞₀𝑡^𝑛𝑒^−𝜆𝑡|𝑟(𝑡,𝛼)|𝑑𝑡‖𝑥‖≤sup_𝑡≥0|𝑟(𝑡,𝛼)|‖𝑥‖. (3.19) 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 (𝑊_𝑎,𝑘,‖⋅‖_𝑎,𝑘) is a Banach space such that 𝑊_𝑎,𝑘⊂𝑋, ‖⋅‖≤‖⋅‖_𝑎,𝑘 and the operator 𝐵_𝑎,𝑘=𝐴|_{𝐷(𝐵𝑎,𝑘)} with 𝐷(𝐵_𝑎,𝑘)∶={𝑥∈𝐷(𝐴)∶𝑥,𝐴𝑥∈𝑊_𝑎,𝑘} generates an (𝑎,𝑘)-regularized family of contractions in 𝑊_𝑎,𝑘, then 𝑊_𝑎,𝑘⊂𝑍_𝑎,𝑘, |⋅|_𝑎,𝑘≤‖⋅‖_𝑎,𝑘 and 𝐵_𝑎,𝑘⊂𝐴_𝑎,𝑘.

Proof. Suppose that (𝑊_𝑎,𝑘,‖⋅‖_𝑎,𝑘), 𝐵_𝑎,𝑘 are as in the statement of theorem. Since 𝐻_𝑎,𝑘(𝜆) is the Laplace transform of the (𝑎,𝑘)-regularized family 𝑆_𝑎,𝑘(𝑡)  (𝑡≥0), we have that for 𝑥∈𝑊_𝑎,𝑘, 𝜆>0, and 𝑛∈ℕ₀1𝜆𝑛!^𝑛+1𝐻^(𝑛)_𝑎,𝑘(𝜆)𝑥=1𝜆𝑛!^𝑛+1𝑑𝑑𝜆^𝑛∞₀𝑒^−𝜆𝑡𝑆_𝑎,𝑘(𝑡)≤1𝑥𝑑𝑡𝜆𝑛!^𝑛+1∞₀𝑒^−𝜆𝑡𝑆_𝑎,𝑘(𝑡)𝑥≤1𝑑𝑡𝜆𝑛!^𝑛+1∞₀𝑒^−𝜆𝑡𝑆_𝑎,𝑘(𝑡)𝑥_𝑎,𝑘≤1𝑑𝑡𝜆𝑛!^𝑛+1∞₀𝑒^−𝜆𝑡‖𝑥‖_𝑎,𝑘=‖𝑥‖𝑑𝑡_𝑎,𝑘. (3.20) We conclude that for 𝑥∈𝑊_𝑎,𝑘, |𝑥|_𝑎,𝑘≤‖𝑥‖_𝑎,𝑘, that is, 𝑊_𝑎,𝑘⊂𝑌_𝑎,𝑘. It follows that 𝐷𝐵_𝑎,𝑘∶=𝑥∈𝐷(𝐴)∶𝑥,𝐴𝑥∈𝑊_𝑎,𝑘⊂𝑥∈𝐷(𝐴)∶𝑥,𝐴𝑥∈𝑌_𝑎,𝑘∶=𝐷𝐴_{𝑌𝑎,𝑘}. (3.21) Hence 𝑊_𝑎,𝑘=𝐷𝐵_𝑎,𝑘^{‖⋅‖𝑎,𝑘}⊂𝐷𝐴_{𝑌𝑎,𝑘}^{‖⋅‖𝑎,𝑘}⊂𝐷𝐵_𝑎,𝑘^{|⋅|𝑎,𝑘}=𝑍_𝑎,𝑘. (3.22) Finally, this implies that 𝐷(𝐵_𝑎,𝑘)⊂𝐷(𝐴_𝑎,𝑘) and 𝐵_𝑎,𝑘⊂𝐴_𝑎,𝑘.

The next result treats 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∈𝐶¹(ℝ₊), 𝑘≠0 with absolutely convergent Laplace transform for 𝜆>0. Assume that 1/𝑎(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.

Proof. By Hille-Yosida theorem it is sufficient to have (𝛼,∞)⊂𝜌(𝐴_𝑎,𝑘) and ‖𝜆(𝜆−𝐴_𝑎,𝑘)‖_{𝐵(𝑍𝑎,𝑘)}≤1 for all 𝜆>𝛼, for some real 𝛼.
In order to show this, we take 𝑛=0 in the second part of Lemma 3.4 and obtain 𝑘(𝜆)𝑎(𝜆)1𝑎(𝜆)−𝐴_𝑎,𝑘⁻¹_{𝐵(𝑍𝑎,𝑘)}≤1𝜆. (3.23) Let 𝜇∶=1/𝑎(𝜆). Then (3.23) gives 𝜇𝜇−𝐴_𝑎,𝑘⁻¹≤1𝜆𝑘(𝜆),(𝜆>0). (3.24) Since 𝑎(𝜆)→0 as 𝜆→∞, and lim_𝜆→∞𝜆𝑘(𝜆)=𝑘(0) by the initial value theorem, we obtain 𝜇𝜇−𝐴_𝑎,𝑘⁻¹_{𝐵(𝑍𝑎,𝑘)}≤1𝑘(0), (3.25) for all 𝜇 sufficiently large. Since 𝑘(0)≥1, we get ‖𝜇(𝜇−𝐴_𝑎,𝑘)‖_{𝐵(𝑍𝑎,𝑘)}≤1 for 𝜇 sufficiently large, which concludes the proof.

Remark 3.9. From the maximal uniqueness of 𝑍_1,1 and under the same hypothesis of preceding theorem we obtain that 𝑍_𝑎,𝑘⊂𝑍_1,1, |⋅|_1,1≤|⋅|_𝑎,𝑘 and 𝐴_𝑎,𝑘⊂𝐴_1,1. Note that, in particular, 𝑍_𝑎,1⊂𝑍_1,1 which include [3, Remark  2.10].

4. Applications

Taking 𝑘(𝑡)≡1 we obtain the main result in [3, Theorem  2.5].

Corollary 4.1 (see [6], Theorem  2.5). Let 𝐴 be a linear operator defined in a Banach space 𝑋 and 𝑎∈𝐿¹_loc(ℝ₊), 𝑎≠0 with absolutely convergent Laplace transform for 𝜆>0. Assume that 1/𝑎(𝜆)∈𝜌(𝐴) for all 𝜆>0. Then there exist a linear subspace 𝑍_𝑎,1 and a norm |⋅|_𝑎,1 such that (𝑍_𝑎,1,|⋅|_𝑎,1) is a Banach space and the equation 𝑢(𝑡)=𝑓(𝑡)+^𝑡₀𝑎(𝑠−𝑡)𝐴_𝑘,1𝑢(𝑠)𝑑𝑠 (4.1) admits resolvent family of contractions on 𝑍_𝑎,1, where 𝐴_𝑘,1 is defined in Lemma 3.4.

Taking 𝑘(𝑡)≡1 or 𝑘(𝑡)≡𝑡, we obtain from the preceding corollary the following.

Corollary 4.2 (see [1]). Let 𝐴 be a linear operator on 𝑋 such that (0,∞)⊂𝜌(𝐴). Then there exist a linear subspace 𝑍₁⊂𝑋 and a norm |⋅|₁ such that (𝑍₁,|⋅|₁) is a Banach space and the restriction 𝐴₁ of 𝐴 to 𝑍₁ is the infinitesimal generator of a 𝐶₀-semigroup of contractions on 𝑍₁.

Corollary 4.3 (see [2]). 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 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)≠∅ 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.

Example 4.5. Let us consider 𝑘(𝑡)∶=𝑡^𝛽/(Γ(𝛽+1)) and 𝑎(𝑡)∶=𝑡^𝛼−1/Γ(𝛼). Let 𝐴 be a closed linear and densely defined operator on a Banach space 𝑋 such that (a)(0,∞)⊂𝜌(𝐴), (b)𝜎_𝑝(𝐴)∩ℝ₋≠∅, (c)0<𝛼<2 and 𝛼≥𝛽.
Let 𝜇∈𝜎_𝑝(𝐴)∩ℝ₋. Then applying Laplace transform, we have that 𝑟̂=𝜆𝜆,𝜇^{𝛼−(𝛽+1)}𝜆^𝛼.−𝜇 (4.2)
It follows from (2.4) that 𝑟𝑡,𝜇=𝑡^𝛽𝐸_𝛼,𝛽+1𝜇𝑡^𝛼,𝜇<0. (4.3)
Take 𝑧=𝜇𝑡^𝛼 with 𝑡>0, then arg(𝑧)=𝜋. By Remark 2.4, for each 𝜋(𝛼/2)<𝜔<min{𝜋,𝜋𝛼}, there is a constant 𝐶∶=𝐶(𝜔)>0 such that 𝐸_𝛼,𝛽+1𝜇𝑡^𝛼≤𝐶1+𝜇𝑡^𝛼,𝑡>0. (4.4) Therefore 𝑟𝑡,𝜇=𝑡^𝛽𝐸_𝛼,𝛽+1𝜇𝑡^𝛼≤𝐶𝑡^𝛽1+𝜇𝑡^𝛼,𝑡>0. (4.5) From here and part (c), it follows that 𝑟(𝑡,𝜇) is a bounded function for 𝑡>0. In particular, 𝑍_𝑎,𝑘≡𝑍_𝛼,𝛽≠{0} by Proposition 3.6. Since (0,∞)⊂𝜌(𝐴), we have that 1/𝑎(𝜆)=𝜆²∈𝜌(𝐴) for 𝜆>0. It follows from Theorem 3.5 that 𝐴 is the infinitesimal generator of a strongly continuous Laplace transformable (𝛼,𝛽)-resolvent family of contractions on 𝑍_𝛼,𝛽. Note in particular that if 𝛼=1 and 𝛽>0, then 𝐴 is the generator of 𝛽-times integrated semigroup (𝑆_𝛽(𝑡))_𝑡≥0 on each subspace 𝑍_1,𝛽 for 0<𝛽≤1. It means that the initial value problem 𝑢^′(𝑡)(𝑡)𝑢(0)=𝐴𝑢,𝑡≥0,=𝑢₀ (4.6) is well posed in the sense that there exists a strongly continuous family of linear operators (𝑆_𝛽(𝑡))_𝑡≥0 on a subspace 𝑍_1,𝛽 of 𝑋 such that for all initial values 𝑢₀∈𝑍_1,𝛽 there exists a unique classical solution of (4.6).

Example 4.6. Let us consider 𝑘(𝑡)∶=𝑡^𝛽/(Γ(𝛽+1)) and 𝑎(𝑡)∶=𝑡^𝛼−1/Γ(𝛼) when 𝛼=2, that is, 𝑎(𝑡)∶=𝑡. Let 𝐴 be a closed linear and densely defined operator on a Banach space 𝑋 such that (a)(0,∞)⊂𝜌(𝐴),(b)𝜎_𝑝(𝐴)∩ℝ₋≠∅, (c)0<𝛽≤1.
Let 𝜇∈𝜎_𝑝(𝐴)∩ℝ₋. Then 𝑟̂=𝜆𝜆,𝜇^1−𝛽𝜆²=𝜆−𝜇𝜆²1−𝜇𝜆^𝛽. (4.7)
Let 𝜔²∶=−𝜇. First, we consider 𝛽=1. In this case, if we take inverse Laplace transform, we obtain 𝑟=1𝑡,𝜇𝜔sin(𝜔𝑡). (4.8)
Obviously, this function is bounded for 𝑡≥0 and 𝜇=−𝜔²<0.
Now, we consider 0<𝛽<1. In this case 𝑟𝑡,𝜇=^𝑡₀𝑠^𝛽−1Γ𝛽(𝜔(=1cos𝑡−𝑠))𝑑𝑠Γ𝛽cos(𝜔𝑡)^𝑡₀𝑠^𝛽−1cos(𝜔𝑠)𝑑𝑠+sin(𝜔𝑡)^𝑡₀𝑠^𝛽−1sin(𝜔𝑠)𝑑𝑠. (4.9)
Remember that the incomplete Gamma function is defined by Γ(𝑎,𝑧)∶=^∞_𝑧𝑒^−𝑡𝑡^𝑎−1𝑑𝑡. (4.10)
About the asymptotic behavior, we know that (see [23, formula (8.357)]) Γ(𝑎,𝑧)∶=𝑧^𝑎−1𝑒^−𝑧1+𝒪|𝑧|⁻¹, (4.11) as |𝑧|→∞ and −3𝜋/2<arg(𝑧)<3𝜋/2.
It can be verified that the following formulas holds (see [23, formula (2.632)]): ^𝑡₀𝑠^𝛽−1()1sin𝜔𝑠𝑑𝑠=−2𝜔^𝛽exp𝑖𝜋2𝛽−1Γ𝛽,−𝑖𝜔𝑡+exp𝑖𝜋21−𝛽Γ𝛽,𝑖𝜔𝑡,𝛽<1_𝑡0,𝑠^𝛽−1()1cos𝜔𝑠𝑑𝑠=−2𝜔^𝛽exp𝑖𝛽𝜋2Γ𝛽,−𝑖𝜔𝑡+exp𝑖𝛽𝜋2Γ𝛽,𝑖𝜔𝑡. (4.12)
Note that arg(𝑖𝜔𝑡)=𝜋/2 and arg(𝑖𝜔𝑡)=−𝜋/2, so we can apply (4.11). Then, for 𝑡 sufficiently large, we obtain that Γ𝛽,𝑖𝜔𝑡=Γ𝛽,−𝑖𝜔𝑡≤𝜔^𝛽−1𝑡^𝛽−11+𝒪|𝜔𝑡|⁻¹. (4.13)
The boundedness follows from the fact that 𝛽<1. Therefore, for 𝑡 sufficiently large 𝑟𝑡,𝜇≤𝑀Γ𝛽,𝑖𝜔𝑡+Γ𝛽,−𝑖𝜔𝑡≤𝐾, (4.14) where 𝐾 depends on 𝜔 and 𝛽. Since (0,∞)⊂𝜌(𝐴), we have that 𝜆²∈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 𝑢^′′(𝑡)(𝑡)𝑢(0)=𝐴𝑢,𝑡≥0,=𝑢₀,𝑢^′(0)=0 (4.15) 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 𝑢₀∈𝑍_2,𝛽 there exists a unique classical solution of (4.15).

Acknowledgments

The first author is partially supported by a research grant of Banco Santander and Universidad Tecnológica de Bolívar. The second author is partially supported by Proyecto FONDECYT 1100485.