(𝑎,𝑘)-Regularized 𝐶-Resolvent Families: Regularity and Local Properties

Abstract

We introduce the class of (local) (𝑎,𝑘)-regularized 𝐶-resolvent families and discuss its basic structural properties. In particular, our analysis covers subjects like regularity, perturbations, duality, spectral properties and subordination principles. We apply our results in the study of the backwards fractional diffusion-wave equation and provide several illustrative examples of differentiable (𝑎,𝑘)-regularized 𝐶-resolvent families.

1. Introduction and Preliminaries

In this review, we will report how a large number of known results concerning (𝑎,𝑘)-regularized resolvents [1–6], 𝐶-regularized resolvents [7], and (local) convoluted 𝐶-semigroups and cosine functions [8, 9] can be formulated in the case of general (𝑎,𝑘)-regularized 𝐶-resolvent families.

The paper is organized as follows. In Theorem 2.2, Remark 2.3, and Theorems 2.5, 2.6, and 2.7, we analyze the properties of subgenerators of (𝑎,𝑘)-regularized 𝐶-resolvent families and slightly improve results from [1]. With a view to further study the problem describing heat conduction in materials with memory and the Rayleigh problem of viscoelasticity in 𝐿^∞ type spaces, we prove in Theorem 2.8 several different forms of subordination principles [10]. The main objective in Theorems 2.9–2.12, 2.26, 2.28, and 2.32 is to continue the researches raised in [3] and [5, 6]. Our main contributions are Theorems 2.16–2.17, 2.20–2.25, 2.27, and 2.30 clarifying the basic regularity properties of (𝑎,𝑘)-regularized 𝐶-resolvent families and a fairly general form of the abstract Weierstrass formula.

It is noteworthy that the complete spectral characterization of subgenerators of (𝑎,𝑘)-regularized 𝐶-resolvent families exists only in the exponential case and that it is not clear, with exception of various types of local convoluted 𝐶-semigroups and cosine functions [9, 11], in what way one can prove a satisfactory Hille-Yosida theorem for local (𝑎,𝑘)-regularized 𝐶-resolvent families.

Throughout this paper 𝐸 denotes a nontrivial complex Banach space, 𝐿(𝐸) denotes the space of boundedlinear operators from 𝐸 into 𝐸,𝐸^∗ denotes the dual space of 𝐸, and 𝐴 denotes a closed linear operator acting on 𝐸. The range and the resolvent set of 𝐴aredenoted by Rang(𝐴) and 𝜌(𝐴), respectively; [𝐷(𝐴)] denotes the Banach space 𝐷(𝐴) equipped with the graph norm. From now on, we assume that 𝐿(𝐸)∋𝐶 is an injective operator which satisfies 𝐶𝐴⊆𝐴𝐶 and employ the convolution like mapping ∗ which is given by 𝑓∗𝑔(𝑡)∶=∫^𝑡₀𝑓(𝑡−𝑠)𝑔(𝑠)𝑑𝑠. Recall, the 𝐶-resolvent set of 𝐴, denoted by 𝜌_𝐶(𝐴), is defined to be the set of all complex numbers 𝜆 satisfying that the operator 𝜆−𝐴 is injective and that Rang(𝐶)⊆Rang(𝜆−𝐴). Let us recall that a linear subspace 𝑌⊆𝐷(𝐴) is called a core for 𝐴 if 𝑌 is dense in 𝐷(𝐴) with respect to the graph norm. Henceforth we identify a closed linear operator 𝐴 with its graph 𝐺(𝐴); given two closed linear operators 𝐴 and 𝐵 on 𝐸, the inclusion 𝐴⊆𝐵 means 𝐺(𝐴)⊆𝐺(𝐵). If 𝑋 is a closed subspace of 𝐸, then 𝐴_𝑋 denotes the part of 𝐴 in 𝑋, that is, 𝐴_𝑋∶={(𝑥,𝑦)∈𝐴∶𝑥∈𝑋,𝑦∈𝑋}.

We mainly use the following conditions.

(H1):A is densely defined.(H2):𝜌(𝐴)≠∅.(H3):𝜌_𝐶(𝐴)≠∅ and Rang(𝐶)=𝐸.(H4):𝐴isdenselydefinedor𝜌_𝐶(𝐴)≠∅.(H5):(H1) ∨ (H2) ∨ (H3).(P1):𝑘(𝑡) is Laplace transformable, that is, it is locally integrable on [0,∞) and there exists 𝛽∈ℝ so that 𝑘^∼(𝜆)=ℒ(𝑘)(𝜆)∶=lim_𝑏→∞∫^𝑏₀𝑒^−𝜆𝑡𝑘(𝑡)𝑑𝑡∶=∫^∞₀𝑒^−𝜆𝑡𝑘(𝑡)𝑑𝑡 exists for all 𝜆∈ℂ with Re𝜆>𝛽. Put abs(𝑘)∶=inf{Re𝜆∶𝑘^∼(𝜆)exists}.

Let us remind that a function 𝑘∈𝐿¹_loc([0,𝜏)) is called a kernel, if for every 𝜙∈𝐶([0,𝜏)), the supposition ∫^𝑡₀𝑘(𝑡−𝑠)𝜙(𝑠)𝑑𝑠=0,𝑡∈[0,𝜏), implies 𝜙≡0; due to the famous Titchmarsh's theorem [12], the condition 0∈ supp?𝑘 implies that 𝑘(𝑡) is a kernel. Set Θ(𝑡)∶=∫^𝑡₀𝑘(𝑠)𝑑𝑠,𝑡∈[0,𝜏).

2. (𝑎,𝑘)-Regularized 𝐶-Resolvent Families

We start with the following definition.

Definition 2.1. Let 0<𝜏≤∞,𝑘∈𝐶([0,𝜏)),𝑘≠0, and let 𝑎∈𝐿¹_loc([0,𝜏)),𝑎≠0. A strongly continuousoperator family (𝑅(𝑡))_{𝑡∈[0,𝜏)} is called a (local, if 𝜏<∞) (𝑎,𝑘)-regularized 𝐶-resolventfamily having 𝐴 as a subgeneratorif and only ifthe following holds: (i)𝑅(𝑡)𝐴⊆𝐴𝑅(𝑡),𝑡∈[0,𝜏),𝑅(0)=𝑘(0)𝐶, and 𝐶𝐴⊆𝐴𝐶,(ii)𝑅(𝑡)𝐶=𝐶𝑅(𝑡),𝑡∈[0,𝜏),(iii)𝑅(𝑡)𝑥=𝑘(𝑡)𝐶𝑥+∫^𝑡₀𝑎(𝑡−𝑠)𝐴𝑅(𝑠)𝑥𝑑𝑠,𝑡∈[0,𝜏),𝑥∈𝐷(𝐴).In the case 𝜏=∞,(𝑅(𝑡))_𝑡≥0 is said to be exponentially bounded if, additionally, there exist 𝑀>0 and 𝜔≥0 such that ‖𝑅(𝑡)‖≤𝑀𝑒^𝜔𝑡,𝑡≥0;(𝑅(𝑡))_{𝑡∈[0,𝜏)} is said to be nondegenerate if the condition 𝑅(𝑡)𝑥=0,𝑡∈[0,𝜏) implies 𝑥=0.
From now on, we consider only nondegenerate (𝑎,𝑘)-regularized 𝐶-resolvent families. Notice that (𝑅(𝑡))_{𝑡∈[0,𝜏)} is nondegenerate provided that 𝑘(0)≠0 or that (H5) holds for a subgenerator 𝐴 of (𝑅(𝑡))_{𝑡∈[0,𝜏)}.
In the case 𝑘(𝑡)=𝑡^𝛼/Γ(𝛼+1), where 𝛼>0, and Γ(⋅) denotes the Gamma function, it is also said that (𝑅(𝑡))_{𝑡∈[0,𝜏)} is an 𝛼-times integrated (𝑎,𝐶)-resolvent family; in such a way, we unify the notion of (local) 𝛼-times integrated 𝐶-semigroups (𝑎(𝑡)≡1) and cosine functions (𝑎(𝑡)≡𝑡) [1, 13, 14]. Furthermore, in the case 𝑘(𝑡)∶=∫^𝑡₀𝐾(𝑠)𝑑𝑠,𝑡∈[0,𝜏), where 𝐾∈𝐿¹_loc([0,𝜏)) and 𝐾≠0, we obtain the unification concept for (local) 𝐾-convoluted 𝐶-semigroups and cosine functions [15]. In the case 𝑘(𝑡)≡1,(𝑅(𝑡))_{𝑡∈[0,𝜏)} is said to be a (local) (𝑎,𝐶)-regularized resolvent family with a subgenerator 𝐴 (cf. also [16] for the definition which does not include the condition (ii) of Definition 2.1).
Designate by ℘(𝑅) the set which consists of all subgenerators of (𝑅(𝑡))_{𝑡∈[0,𝜏)}.
Then the following holds.
(i)𝐴∈℘(𝑅) implies 𝐶⁻¹𝐴𝐶∈℘(𝑅).(ii)If 𝐴∈℘(𝑅) and 𝜆∈𝜌_𝐶(𝐴), then 𝑅(𝑡)(𝜆−𝐴)⁻¹𝐶=(𝜆−𝐴)⁻¹𝐶𝑅(𝑡),𝑡∈[0,𝜏). (2.1)(iii)Assume, additionally, that 𝑎(𝑡) is a kernel. Then one can define the integral generator 𝐴 of (𝑅(𝑡))_{𝑡∈[0,𝜏)} by setting 𝐴∶={(𝑥,𝑦)∈𝐸×𝐸∶𝑅(𝑡)𝑥−𝑘(𝑡)𝐶𝑥=^𝑡₀𝑎(𝑡−𝑠)𝑅(𝑠)𝑦𝑑𝑠,𝑡∈[0,𝜏)}. (2.2) The integral generator 𝐴 of (𝑅(𝑡))_{𝑡∈[0,𝜏)} is a closed linear operator which satisfies 𝐶⁻¹𝐴𝐶=𝐴 and extends an arbitrary subgenerator of (𝑅(𝑡))_{𝑡∈[0,𝜏)}. Furthermore, 𝐴∈℘(𝑅), if 𝑅(𝑡)𝑅(𝑠)=𝑅(𝑠)𝑅(𝑡),0≤𝑡,𝑠<𝜏.
Recall that in the case of convoluted 𝐶-semigroups and cosine functions, the set ℘(𝑅) becomes a complete lattice under suitable algebraic operations and that induced partial ordering coincides with the usual set inclusion. In general, ℘(𝑅) needs not to be finite [9].
Henceforth we assume that the scalar-valued kernels 𝑘,𝑘₁,𝑘₂,… are continuous on [0,𝜏), and that 𝑎≠0 in 𝐿¹_loc([0,𝜏)).
Assume temporarily 𝜆∈𝜌_𝐶(𝐴),𝑥∈Rang(𝐶),𝑡∈[0,𝜏), and put 𝑧=(𝑎∗𝑅)(𝑡)𝑥.
Following the proof of [1, Lemma ?2.2], we have 𝑧=𝜆(𝑎∗𝑅)(𝑡)(𝜆−𝐴)⁻¹𝑥−(𝑎∗𝑅)(𝑡)𝐴(𝜆−𝐴)⁻¹𝑥=𝜆(𝑎∗𝑅)(𝑡)(𝜆−𝐴)⁻¹𝑥−(𝑅(𝑡)(𝜆−𝐴)⁻¹𝑥−𝑘(𝑡)𝐶(𝜆−𝐴)⁻¹𝑥)=𝜆(𝜆−𝐴)⁻¹𝐶(𝑎∗𝑅)(𝑡)𝐶⁻¹𝑥−((𝜆−𝐴)⁻¹𝑅(𝑡)𝑥−𝑘(𝑡)(𝜆−𝐴)⁻¹𝐶𝑥), where the last two equalities follow on account of 𝐶𝐴⊆𝐴𝐶,𝑅(𝑠)𝐴⊆𝐴𝑅(𝑠) and 𝑅(𝑠)(𝜆−𝐴)⁻¹𝐶=(𝜆−𝐴)⁻¹𝐶𝑅(𝑠),𝑠∈[0,𝜏). Hence, (𝜆−𝐴)𝑧=𝜆𝑧−(𝑅(𝑡)𝑥−𝐶𝑥),
^𝑡₀𝑎(𝑡−𝑠)𝑅(𝑠)𝑥𝑑𝑠∈𝐷(𝐴),𝐴^𝑡₀𝑎(𝑡−𝑠)𝑅(𝑠)𝑥𝑑𝑠=𝑅(𝑡)𝑥−𝑘(𝑡)𝐶𝑥. (2.3)
The closedness of 𝐴 implies that (2.3) holds for every 𝑡∈[0,𝜏) and 𝑥∈Rang(𝐶).

Theorem 2.2 (see [1]). (i) Let 𝐴 be a subgenerator of an (𝑎,𝑘)-regularized C-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)}, and let (H5) hold. Then (2.3) holds for every 𝑡∈[0,𝜏) and 𝑥∈𝐸. If 𝜌_𝐶(𝐴)≠∅, then (2.3) holds for every t∈[0,𝜏) and 𝑥∈Rang(𝐶).
(ii) Let 𝐴 be a subgenerator of an (𝑎,𝑘_𝑖)-regularized 𝐶-resolvent family (𝑅_𝑖(𝑡))_{𝑡∈[0,𝜏)},𝑖=1,2. Then (𝑘₂∗𝑅₁)(𝑡)=(𝑘₁∗𝑅₂)(𝑡),𝑡∈[0,𝜏), whenever (H4) holds.
(iii) Let (𝑅₁(𝑡))_{𝑡∈[0,𝜏)} and (𝑅₂(𝑡))_{𝑡∈[0,𝜏)} be two (𝑎,𝑘)-regularized 𝐶-resolvent families having 𝐴 as a subgenerator. Then 𝑅₁(𝑡)𝑥=𝑅₂(𝑡)𝑥,𝑡∈[0,𝜏),𝑥∈𝐷(𝐴), and 𝑅₁(𝑡)=𝑅₂(𝑡),𝑡∈[0,𝜏), if (H4) holds.
(iv) Let 𝐴 be a subgenerator of an (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)}. If 𝑘(𝑡) is absolutely continuous and 𝑘(0)≠0, then 𝐴 is a subgenerator of an (𝑎,𝐶)-regularized resolvent family on [0,𝜏).

Remark 2.3. (i) Let (𝑅_𝑖(𝑡))_{𝑡∈[0,𝜏)} be an (𝑎,𝑘_𝑖)-regularized 𝐶-resolvent family with a subgenerator 𝐴, ??𝑖=1,2, and let 𝐷(𝐴)≠{0}. Then 𝑘₁=𝑘₂.
(ii) Let (𝑅_𝑖(𝑡))_{𝑡∈[0,𝜏)} be an (𝑎,𝑘_𝑖)-regularized 𝐶-resolvent family with a subgenerator 𝐴,𝑖=1,2. Then, for every 𝛼∈ℂ and 𝛽∈ℂ,(𝛼𝑅₁(𝑡)+𝛽𝑅₂(𝑡))_{𝑡∈[0,𝜏)} is an (𝑎,𝛼𝑘₁+𝛽𝑘₂)-regularized 𝐶-resolvent family with a subgenerator 𝐴.
(iii) Let (𝑅(𝑡))_{𝑡∈[0,𝜏)} be an (𝑎,𝑘)-regularized 𝐶-resolvent family with a subgenerator 𝐴, and let 𝐿¹_loc([0,𝜏))∋𝑏 be a kernel. Then 𝐴 is a subgenerator of an (𝑎,𝑘∗𝑏)-regularized 𝐶-resolvent family ((𝑏∗𝑅)(𝑡))_{𝑡∈[0,𝜏)}.
(iv) Let (𝑅(𝑡))_{𝑡∈[0,𝜏)} be an (𝑎,𝐶)-regularized resolvent family having 𝐴 as a subgenerator. Then ((𝑘∗𝑅)(𝑡))_{𝑡∈[0,𝜏)} is an (𝑎,Θ)-regularized 𝐶-resolvent family with a subgenerator 𝐴.
(v) Suppose (𝑅(𝑡))_{𝑡∈[0,𝜏)} is an (𝑎,𝑘)-regularized 𝐶-resolvent family with a subgenerator 𝐴, (H1) or (H3) holds, and 𝑎(𝑡) is a kernel. Then the integral generator 𝐴 of (𝑅(𝑡))_{𝑡∈[0,𝜏)} satisfies 𝐴=𝐶⁻¹𝐴𝐶. Toward this end, let (𝑥,𝑦)∈𝐴. Then ∫^𝑡₀𝑎(𝑡−𝑠)[𝑘(𝑠)𝐶𝑥+∫^𝑠₀𝑎(𝑠−𝑟)𝑅(𝑟)𝑦𝑑𝑟]𝑑𝑠=∫^𝑡₀𝑎(𝑡−𝑠)𝑅(𝑠)𝑥𝑑𝑠∈𝐷(𝐴),𝑡∈[0,𝜏), and 𝐴∫^𝑡₀𝑎(𝑡−𝑠)[𝑘(𝑠)𝐶𝑥+∫^𝑠₀𝑎(𝑠−𝑟)𝑅(𝑟)𝑦𝑑𝑟]𝑑𝑠=𝐴∫^𝑡₀𝑎(𝑡−𝑠)𝑅(𝑠)𝑥𝑑𝑠=𝑅(𝑡)𝑥−𝑘(𝑡)𝐶𝑥=∫^𝑡₀𝑎(𝑡−𝑠)𝑅(𝑠)𝑦𝑑𝑠,𝑡∈[0,𝜏). Since (𝑎∗𝑅)(𝑡)𝑦∈𝐷(𝐴),(𝑎∗𝑎∗𝑅)(𝑡)𝑦∈𝐷(𝐴),𝐴(𝑎∗𝑎∗𝑅)(𝑡)𝑦=(𝑎∗(𝑅−𝑘𝐶))(𝑡)𝑦,𝑡∈[0,𝜏), and 𝑎∗𝑘≠0 in 𝐶([0,𝜏)), it follows that 𝐶𝑥∈𝐷(𝐴),𝐴𝐶𝑥=𝐶𝑦,𝑥∈𝐷(𝐶⁻¹𝐴𝐶), and 𝐶⁻¹𝐴𝐶𝑥=𝐴𝑥=𝑦. On the other hand, 𝐶⁻¹𝐴𝐶 is a subgenerator of (𝑅(𝑡))_{𝑡∈[0,𝜏)} whenever 𝐴 is; this implies 𝐶⁻¹𝐴𝐶⊆𝐴 and proves the claim. If (H2) holds, then 𝐴=𝐶⁻¹𝐴𝐶=𝐴. In what follows, we also assume that 𝐵∈℘(𝑅) and that (H5) holds for 𝐵 and 𝐶. Proceeding as in the proof of [9, Proposition ?2.1.1.6], one gets what follows.
(v.1)𝐶⁻¹𝐴𝐶=𝐶⁻¹𝐵𝐶 and 𝐶(𝐷(𝐴))⊆𝐷(𝐵).(v.2)𝐴 and 𝐵 have the same eigenvalues.(v.3)The assumption 𝐴⊆𝐵 implies 𝜌_𝐶(𝐴)⊆𝜌_𝐶(𝐵).(v.4)card (℘(𝑅))=1, if 𝐶(𝐷(𝐴)) is a core for 𝐷(𝐴).(v.5)𝐴⊆𝐵⇔𝐷(𝐴)⊆𝐷(𝐵) and 𝐴𝑥=𝐵𝑥,𝑥∈𝐷(𝐴)∩𝐷(𝐵); furthermore, the property (v.5) holds whenever {𝐴,𝐵}⊆℘(𝑅) and 𝑎(𝑡) is a kernel.
We refer the reader to [1, page 283] for the definition of (weak) solutions of the problem
𝑢(𝑡)=𝑓(𝑡)+^𝑡₀𝑎(𝑡−𝑠)𝐴𝑢(𝑠)𝑑𝑠,𝑡∈[0,𝜏), (2.4) where 𝑓∈𝐶([0,𝜏)∶𝐸), and to [1, page 285] for the notion of spaces 𝐶^𝑛,𝑘([0,𝜏)∶𝐸),𝑛∈ℕ,𝑘∈ℕ₀ and 𝐶^𝑛₀([0,𝜏)∶𝐸),and𝑛∈ℕ.
Define a subset 𝐴^∗ of 𝐸^∗×𝐸^∗ (the use of symbol ∗ is clear from the context) by 𝐴^∗∶={(𝑥^∗,𝑦^∗)∈𝐸^∗×𝐸^∗∶𝑥^∗(𝐴𝑥)=𝑦^∗(𝑥)forall𝑥∈𝐷(𝐴)}. In the case when 𝐴 is densely defined, 𝐴^∗ is a linear mapping from 𝐸^∗ into 𝐸^∗.

Lemma 2.4 (see [17]). Let 𝐴 be a closed linear operator. Assume 𝑥₀∈𝐸,𝑦₀∈𝐸, and 𝑥^∗(𝑦₀)=𝑦^∗(𝑥₀) for all (𝑥^∗,𝑦^∗)∈𝐴^∗. Then 𝑥₀∈𝐷(𝐴), and 𝐴𝑥₀=𝑦₀.

Define the mapping 𝐾_𝐶∶𝐶([0,𝜏)∶𝐸)→𝐶([0,𝜏)∶𝐸) by 𝐾_𝐶𝑢∶=𝑘∗𝐶𝑢,𝑢∈𝐶([0,𝜏)∶𝐸). Then 𝐾_𝐶 is linear, bounded, and injective.

Keeping in mind Lemma 2.4 and the proofs of [1, Theorem ?2.7, Corollary ?2.9, Remark ?2.10, Corollary ?2.11, and Corollary ??2.13], we have the following.

Theorem 2.5. (i) Suppose 𝑓∈𝐶([0,𝜏)∶𝐸),𝐴 is a subgenerator of a (local) (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)}, and (H5) holds. Then (2.4) has a unique solution if and only if 𝑅∗𝑓∈Rang(𝐾_𝐶).
(ii) (cf. also [18]) Assume 𝑛∈ℕ,𝑓∈𝐶([0,𝜏)∶𝐸),𝐴 is a subgenerator of a (local) 𝑛-times integrated (𝑎,𝐶)-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)}, and (H5) holds. Then (2.4) has a unique solution if and only if 𝐶⁻¹(𝑅∗𝑓)∈𝐶₀^𝑛+1([0,𝜏)∶𝐸).
(iii) Let the assumptions of the item (i) of this theorem hold, and let 𝑘∈𝐶^𝑛₀([0,𝜏)∶𝐸). Then 𝐶⁻¹(𝑅∗𝑓)∈𝐶^(𝑛+1)([0,𝜏)∶𝐸) if and only if 𝐶⁻¹(𝑅∗𝑓)∈𝐶₀^𝑛+1([0,𝜏)∶𝐸).
(iv) Let (H5) hold. Assume that 𝑛∈ℕ,𝐴 is a subgenerator of an 𝑛-times integrated (𝑎,𝐶)-regularized resolvent, and 𝑎∈𝐵𝑉_loc([0,𝜏)∶𝐸), respectively 𝐴 is a subgenerator of an (𝑎,𝐶)-regularized resolvent family. Assume, further, that 𝐶⁻¹𝑓∈𝐶^(𝑛+1)([0,𝜏)∶𝐸),𝑓^(𝑘−1)(0)∈𝐷(𝐴^{𝑛+1−𝑘}) and 𝐴^{𝑛+1−𝑘}𝑓^(𝑘−1)(0)∈Rang(𝐶),1≤𝑘≤𝑛+1, respectively 𝐶⁻¹𝑓∈𝐴𝐶_loc([0,𝜏)∶𝐸). Then (2.4) has a unique solution.
(v) Assume that (H5) holds, 𝐴 is a subgenerator of an (𝑎,𝑘)-regularized 𝐶-resolvent family, 𝑘(𝑡) is absolutely continuous, and 𝑘(0)≠0. If 𝐶⁻¹𝑓∈𝐶¹([0,𝜏)∶𝐸), then there exists a unique solution of (2.4).

The proof of following theorem follows from a standard application of Laplace transform techniques.

Theorem 2.6. Let 𝑘(𝑡) and 𝑎(𝑡) satisfy (P1), and let (𝑅(𝑡))_𝑡≥0 be a strongly continuous operator family satisfying ‖𝑅(𝑡)‖≤𝑀𝑒^𝜔𝑡,𝑡≥0, for some 𝑀>0 and 𝜔≥0. Put 𝜔₀∶=max(𝜔,abs(𝑎),abs(𝑘)).(i)Suppose 𝐴 is a subgenerator of the exponentially bounded (a,k)-regularized 𝐶-resolvent family (𝑅(𝑡))_𝑡≥0, and (H5) holds. Then, for every 𝜆∈ℂ with Re𝜆>𝜔₀ and 𝑘^∼(𝜆)≠0, the operator 𝐼−𝑎^∼(𝜆)𝐴 is injective, Rang(𝐶)⊆Rang(𝐼−𝑎^∼(𝜆)𝐴),𝑘^∼(𝜆)(𝐼−𝑎^∼(𝜆)𝐴)⁻¹𝐶𝑥=^∞₀𝑒^−𝜆𝑡𝑅(𝑡)𝑥𝑑𝑡,𝑥∈𝐸,Re𝜆>𝜔₀,𝑘^∼(𝜆)≠0, (2.5) ? {_∼1𝑎(𝜆)∶𝜆∈ℂ,Re𝜆>𝜔₀,𝑘^∼(𝜆)𝑎^∼(𝜆)≠0}⊆𝜌_𝐶(𝐴) (2.6) and 𝑅(𝑠)𝑅(𝑡)=𝑅(𝑡)𝑅(𝑠),𝑡,𝑠≥0.(ii)Assume that (2.5)-(2.6) hold. Then 𝐴 is a subgenerator of the exponentially bounded (𝑎,𝑘)-regularized C-resolvent family (𝑅(𝑡))_𝑡≥0.

The preceding theorem enables one to establish the real and complex characterization of subgenerators of (locally Lipschitz continuous) exponentially bounded (𝑎,𝑘)-regularized 𝐶-resolvent families [1, 9, 12]:

Theorem 2.7. (i) Let 𝑘(𝑡) and 𝑎(𝑡) satisfy (P1), and let 𝜔₀≥max(0,abs(𝑎),abs(𝑘)). Assume that, for every 𝜆∈ℂ with Re𝜆>𝜔₀ and 𝑘^∼(𝜆)≠0, the operator 𝐼−𝑎^∼(𝜆)𝐴 is injective and that Rang(𝐶)⊆Rang(𝐼−𝑎^∼(𝜆)𝐴). If there exists an analytic function Υ∶{𝜆∈ℂ∶Re𝜆>𝜔₀}→𝐿(𝐸) with:
(i.1)Υ(𝜆)=𝑘^∼(𝜆)(𝐼−𝑎^∼(𝜆)𝐴)⁻¹𝐶,𝜆∈ℂ,Re𝜆>𝜔₀, (i.2)‖Υ(𝜆)‖≤𝑀|𝜆|^𝑟,𝜆∈ℂ,Re𝜆>𝜔₀, for some 𝑀>0 and 𝑟≥−1, then, for every 𝛼>1,𝐴 is a subgenerator of a norm continuous, exponentially bounded (𝑎,𝑘∗𝑡^{𝛼+𝑟−1}/Γ(𝛼+𝑟))-regularized 𝐶-resolvent family.
(ii) Suppose 𝑘(𝑡) and 𝑎(𝑡) satisfy (P1) and (H2) or (H3) holds, and 𝐴 is a subgenerator of an exponentially bounded (𝑎,Θ)-regularized 𝐶-resolvent family (𝑅(𝑡))_𝑡≥0 which satisfies the next condition:
‖𝑅(𝑡+ℎ)−𝑅(𝑡)‖≤𝑀ℎ𝑒^{𝜔(𝑡+ℎ)},𝑡≥0,ℎ≥0,forsome𝑀>0,𝜔≥0. (2.7) Then there exists 𝑎≥𝜔₀ such that {_∼1𝑎(𝜆)∶𝜆>𝑎,𝑘^∼(𝜆)𝑎^∼(𝜆)≠0}⊆𝜌_𝐶(𝐴),themapping𝜆↦𝐻(𝜆)∶=𝑘^∼(𝜆)(𝐼−𝑎^∼(𝜆)𝐴)⁻¹𝐶,𝜆>𝑎,𝑘^∼(𝜆)𝑎^∼(𝜆)≠0 (2.8)(2.9)
(ii) Suppose 𝐴 is a subgenerator of an (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)} satisfying ‖𝑅(𝑡)‖=𝑂(𝑘(𝑡)),𝑡→0+,min(𝑎(𝑡),𝑘(𝑡))>0,𝑡∈(0,𝜏), and (H5) holds. Then (2.18)-(2.19) hold.

Theorem 2.10. (i) Suppose 𝐴 is a subgenerator of an (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)} satisfying ‖𝑅(𝑡)‖=𝑂(𝑘(𝑡)),𝑡→0+ and min(𝑎(𝑡),𝑘(𝑡))>0,𝑡∈(0,𝜏). Then lim_𝑡→0+(𝑎∗𝑅)(𝑡)𝑥/(𝑎∗𝑘)(𝑡)=𝐶𝑥,𝑥∈𝐷(𝐴).
(ii) Suppose 𝐴 is a subgenerator of an (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)} satisfying ‖𝑅(𝑡)‖=𝑂(𝑘(𝑡)),𝑡→0+,min(𝑎(𝑡),𝑘(𝑡))>0,𝑡∈(0,𝜏) and (H5) holds. If 𝑥∈𝐷(𝐴),𝑦∈𝐸 and lim_𝑡→0+(𝑅(𝑡)𝑥−𝑘(𝑡)𝐶𝑥)/(𝑎∗𝑘)(𝑡)=𝑦, then 𝑥∈𝐷(𝐴) and 𝑦=𝐴𝑥.
(iii) Suppose 𝐸 is reflexive, 𝐴 is a subgenerator of an (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)} satisfying ‖𝑅(𝑡)‖=𝑂(𝑘(𝑡)),𝑡→0+,𝑅(𝑠)𝑅(𝑡)=𝑅(𝑡)𝑅(𝑠),0≤𝑡,𝑠<𝜏,min(𝑎(𝑡),𝑘(𝑡))>0,𝑡∈(0,𝜏), and (H5) holds. If 𝑥∈𝐷(𝐴) and lim_𝑡→0+‖(𝑅(𝑡)𝑥−𝑘(𝑡)𝐶𝑥)/(𝑎∗𝑘)(𝑡)‖<∞, then 𝑥∈𝐷(𝐴).

Theorem 2.11 (cf. also [22]). Suppose 𝛼>0 and 𝐴 is a subgenerator of an 𝛼-times integrated 𝐶-semigroup (𝑆_𝛼(𝑡))_{𝑡∈[0,𝜏)}, respectively, an 𝛼-times integrated 𝐶-cosine function (𝐶_𝛼(𝑡))_{𝑡∈[0,𝜏)}, which satisfies limsup_𝑡→0+‖𝑆_𝛼(𝑡)‖/𝑡^𝛼<∞, respectively, limsup_𝑡→0+‖𝐶_𝛼(𝑡)‖/𝑡^𝛼<∞. Then, for every 𝑥∈𝐷(𝐴) such that 𝐴𝑥∈𝐷(𝐴)∶𝐴𝑥=lim_𝑡→0+Γ(𝛼+2)Γ(𝛼+1)Γ(𝛼+1)𝑆_𝛼(𝑡)𝑥−𝑡^𝛼𝐶𝑥𝑡^𝛼+1(resp.),𝐴𝑥=lim_𝑡→0+Γ(𝛼+3)Γ(𝛼+1)Γ(𝛼+1)𝐶_𝛼(𝑡)𝑥−𝑡^𝛼𝐶𝑥𝑡^𝛼+2. (2.20)(2.21) then 𝐴 is a subgenerator of an exponentially bounded, analytic (𝑎,𝑘)-regularized 𝐶-resolvent family of angle 𝛼.

Example 2.18 (cf. also [9, Theorem ?2.1.4.7]). Let 𝛽∈(0,2),𝛼>0,𝑘(𝑡)=𝑡^𝛼/Γ(𝛼+1), and let 𝑎(𝑡)=𝑡^𝛽−1/Γ(𝛽). Let 𝐴 be densely defined. Then 𝐴 is a subgenerator of an exponentially bounded, analytic (𝑎,𝑘)-regularized 𝐶-resolvent family of angle 𝛾 if and only if for every 𝛿∈(0,𝛾), there exist 𝑀_𝛿>0 and 𝜔_𝛿≥0 such that (𝜔_𝛿+Σ_𝜋/2+𝛿)^1/𝛽⊆𝜌_𝐶(𝐴),‖(𝜆^𝛽−𝐴)⁻¹𝐶‖≤𝑀_𝛿(1+|𝜆|)^𝛼−𝛽,𝜆∈(𝜔_𝛿+Σ_𝜋/2+𝛿)^1/𝛽, (2.34) the mapping 𝜆↦(𝜆^𝛽−𝐴)⁻¹𝐶,𝜆∈(𝜔_𝛿+Σ_𝜋/2+𝛿)^1/𝛽 is analytic (continuous).
Let (𝑀_𝑝) be a sequence of positive real numbers such that 𝑀₀=1 and that
(M.1)𝑀²_𝑝≤𝑀_𝑝+1𝑀_𝑝−1,𝑝∈ℕ,(M.2)𝑀_𝑛≤𝐴𝐻^𝑛min_{𝑝,𝑞∈ℕ,𝑝+𝑞=𝑛}𝑀_𝑝𝑀_𝑞,𝑛∈ℕ,forsome𝐴>1,and𝐻>1,(M.3) ∑^∞_𝑝=1(𝑀_𝑝−1/𝑀_𝑝)<∞.
The Gevrey sequences (𝑝!^𝑠),(𝑝^𝑝𝑠), and (Γ(1+𝑝𝑠)) satisfy the above conditions, where 𝑠>1. Put 𝑚_𝑝∶=(𝑀_𝑝/𝑀_𝑝−1),𝑝∈ℕ; by (M.1), (𝑚_𝑝) is increasing, and (M.3)^′ implies ^∞∑_𝑝=1(1/𝑚_𝑝)<∞. The associated function of (𝑀_𝑝) is defined by 𝑀(𝜆)∶=sup_𝑝∈ℕ0ln(|𝜆|^𝑝/𝑀_𝑝),𝜆∈ℂ⧵{0},𝑀(0)∶=0. As is known, the function 𝑡↦𝑀(𝑡),𝑡≥0, is increasing, absolutely continuous, lim_𝑡→∞𝑀(𝑡)=+∞ and lim_𝑡→∞(𝑀(𝑡)/𝑡)=0. For consistency of terminology with [24], we also employ the sequence (𝐿_𝑝∶=𝑀_𝑝^1/𝑝) and set 𝜔_𝐿(𝑡)∶=∑^∞_𝑝=0(𝑡^𝑝/𝐿^𝑝_𝑝),𝑡≥0.
We need the following family of kernels. Define, for every 𝑙>0, the next entire function of exponential type zero 𝜔_𝑙(𝜆)∶=∏^∞_𝑝=11+𝑙𝜆/𝑚_𝑝,𝜆∈ℂ. Then
|𝜔_𝑙(𝜆)|≥sup^𝑘_{𝑘∈ℕ𝑝=1}|1+(𝑙𝜆/𝑚_𝑝)|≥sup^𝑘_{𝑘∈ℕ𝑝=1}𝑙|𝜆|𝑚_𝑝≥sup_𝑘∈ℕ(𝑙|𝜆|)^𝑘𝑀_𝑝,𝜆∈ℂ,Re𝜆≥0, (2.35) and this implies that |𝜔_𝑙(𝜆)|≥𝑒^{𝑀(𝑙|𝜆|)},𝜆∈ℂ,Re𝜆≥0. It is noteworthy that, for every 𝛼∈(0,𝜋/2),𝑝∈ℕ₀ and 𝜆∈Σ_𝜋/2+𝛼,|1+𝑙𝜆/𝑚_𝑝|≥𝑙|𝐼𝑚𝜆|/𝑚_𝑝≥𝑙(1+tan𝛼)⁻¹|𝜆|/𝑚_𝑝.
This yields
|𝜔_𝑙(𝜆)|≥𝑒^{𝑀(𝑙(1+tan𝛼)−1|𝜆|)}𝜋,𝛼∈(0,2),𝑙>0,𝜆∈Σ_{(𝜋/2)+𝛼}. (2.36) Put now 𝑘_𝑙(𝑡)∶=ℒ⁻¹(1𝜔_𝑙(𝜆))(𝑡),𝑡≥0,𝑙>0. (2.37) Then, for every 𝑙>0,0∈supp𝑘_𝑙 and 𝑘_𝑙 is infinitely differentiable in 𝑡≥0.

Definition 2.19. Let (𝑅(𝑡))_{𝑡∈[0,𝜏)} be a (local) (𝑎,𝑘)-regularized 𝐶-resolvent family having 𝐴 as a subgenerator, and let the mapping 𝑡↦𝑅(𝑡),𝑡∈(0,𝜏), be infinitely differentiable (in the uniform operator topology). Then it is said that (𝑅(𝑡))_{𝑡∈[0,𝜏)} is of class 𝐶^𝐿, resp. of class 𝐶_𝐿, if and only if for every compact set 𝐾⊆(0,𝜏) there exists ℎ_𝐾>0, resp. for every compact set 𝐾⊆(0,𝜏) and for every ℎ>0: sup_{𝑡∈𝐾,𝑝∈ℕ0}‖ℎ^𝑝_𝐾(𝑑^𝑝/𝑑𝑡^𝑝)𝑅(𝑡)𝐿^𝑝_𝑝‖<∞,(resp.,)sup_{𝑡∈𝐾,𝑝∈ℕ0}‖ℎ^𝑝(𝑑^𝑝/𝑑𝑡^𝑝)𝑅(𝑡)𝐿^𝑝_𝑝‖<∞, (2.38)(𝑅(𝑡))_{𝑡∈[0,𝜏)} is said to be 𝜌-hypoanalytic, 1≤𝜌<∞, if (𝑅(𝑡))_{𝑡∈[0,𝜏)} is of class 𝐶^𝐿 with 𝐿_𝑝=𝑝!^𝜌/𝑝.

By the proof of the scalar-valued version of the Pringsheim theorem, it follows that the mapping 𝑡↦𝑅(𝑡),𝑡∈(0,𝜏) is real analytic if and only if (𝑅(𝑡))_{𝑡∈[0,𝜏)} is 𝜌-hypoanalytic with 𝜌=1.

The main objective in Theorems 2.20–2.24 is to enquire into the basic differential properties of (𝑎,𝑘)-regularized 𝐶-resolvent families.

Theorem 2.20 ([25]). Suppose𝐴 is a closed linear operator, 𝑘(𝑡) and 𝑎(𝑡) satisfy (P1), 𝑟≥−1, and there exists 𝜔≥max(0,abs(𝑘),abs(𝑎)) such that, for every 𝑧∈{𝜆∈ℂ∶Re𝜆>𝜔,𝑘^∼(𝜆)≠0}, we have that the operator 𝐼−𝑎^∼(𝑧)𝐴 is injective and that Rang(𝐶)⊆Rang(𝐼−𝑎^∼(𝑧)𝐴). If, additionally, for every 𝜎>0, there exist 𝐶_𝜎>0,𝑀_𝜎>0 and an open neighborhood Ω_𝜎,𝜔 of the region Λ_𝜎,𝜔∶={𝜆∈ℂ∶Re𝜆≤𝜔,Re𝜆≥−𝜎ln|Im𝜆|+𝐶_𝜎}∪{𝜆∈ℂ∶Re𝜆≥𝜔}, (2.39) and an analytic mapping ℎ_𝜎∶Ω_𝜎,𝜔→𝐿(𝐸) such that ℎ_𝜎(𝜆)=𝑘^∼(𝜆)(𝐼−𝑎^∼(𝜆)𝐴)⁻¹𝐶,Re𝜆>𝜔,𝑘^∼(𝜆)≠0 and that ‖ℎ_𝜎(𝜆)‖≤𝑀_𝜎|𝜆|^𝑟𝜆∈Λ_𝜎,𝜔, then, for every 𝜁>1,𝐴 is a subgenerator of a norm continuous, exponentially bounded (𝑎,𝑘∗𝑡^{𝜁+𝑟−1}/Γ(𝜁+𝑟))-regularized 𝐶-resolvent family (𝑅(𝑡))_𝑡≥0 satisfying that the mapping 𝑡↦𝑅(𝑡),𝑡>0 is infinitely differentiable.

Theorem 2.21. Suppose 𝑘(𝑡) and 𝑎(𝑡) satisfy (P1), (H5) hold and 𝐴 is a subgenerator of an (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_𝑡≥0 satisfying ‖𝑅(𝑡)‖≤𝑀𝑒^𝜔′𝑡,𝑡≥0 for appropriate constants 𝜔^′≥max(0,abs(𝑘),abs(𝑎)), and 𝑀>0. If there exists 𝜔>𝜔^′ such that, for every 𝜎>0, there exist 𝐶_𝜎>0 and 𝑀_𝜎>0 so that (i)there exist an open neighborhood Ω_𝜎,𝜔 of the region Λ_𝜎,𝜔, and the analytic mappings 𝑓_𝜎∶Ω_𝜎,𝜔→ℂ,𝑔_𝜎∶Ω_𝜎,𝜔→ℂ, and ℎ_𝜎∶Ω_𝜎,𝜔→𝐿(𝐸) such that 𝑓_𝜎(𝜆)=𝑘^∼(𝜆),𝜆∈ℂ,Re𝜆≥𝜔 and 𝑔_𝜎(𝜆)=𝑎^∼(𝜆),𝜆∈ℂ,Re𝜆≥𝜔,(ii)for every 𝜆∈Λ_𝜎,𝜔 with Re𝜆≤𝜔, the operator 𝐼−𝑎^∼(𝜆)𝐴 is injective and Rang(𝐶)⊆Rang(𝐼−𝑎^∼(𝜆)𝐴),(iii)ℎ_𝜎(𝜆)=𝑓_𝜎(𝜆)(𝐼−𝑔_𝜎(𝜆)𝐴)⁻¹𝐶,𝜆∈Λ_𝜎,𝜔,(iv)‖ℎ_𝜎(𝜆)‖≤𝑀_𝜎|Im𝜆|,𝜆∈Λ_𝜎,𝜔,Re𝜆≤𝜔, and max(|𝑓_𝜎(𝜆)|,|𝑔_𝜎(𝜆)|)≤𝑀_𝜎,𝜆∈Λ_𝜎,𝜔,then the mapping 𝑡↦𝑅(𝑡)𝑥,𝑡>0 is infinitely differentiable for every fixed 𝑥∈𝐷(𝐴²). Furthermore, if 𝐷(𝐴²) is dense in 𝐸, then the mapping 𝑡↦𝑅(𝑡),𝑡>0, is infinitely differentiable.

Proof. Assume 𝜎>0,𝜍>0,𝜔₀>𝜔, and put Γ¹_𝜎∶={𝜆∈ℂ∶Re𝜆=2𝐶_𝜎−𝜎ln(−Im𝜆),−∞<Im𝜆≤−𝑒^{(2𝐶𝜎/𝜎)}},Γ²_𝜎∶={𝜆∈ℂ∶Re𝜆=𝜔₀,−𝑒^{(2𝐶𝜎/𝜎)}≤Im𝜆≤𝑒^{(2𝐶𝜎/𝜎)}},Γ³_𝜎∶={𝜆∈ℂ∶Re𝜆=2𝐶_𝜎−𝜎ln(Im𝜆),𝑒^{(2𝐶𝜎/𝜎)}≤Im𝜆<+∞},Γ_𝜎∶=Γ¹_𝜎∪Γ²_𝜎∪Γ³_𝜎, and Γ_𝑘,𝜎∶={𝜆∈Γ_𝜎∶|𝜆|≤𝑘},𝑘∈ℕ. The curves Γ_𝜎 and Γ_𝑘,𝜎 are oriented so that Im𝜆 increases along Γ_𝜎 and Γ_𝑘,𝜎,𝑘∈ℕ. Set, for a sufficiently large 𝑘₀∈ℕ,𝑆^𝑘_𝜎(𝑡)∶=(1/2𝜋𝑖)∫_Γ𝑘,𝜎𝑒^𝜆𝑡(ℎ_𝜎(𝜆)/𝜆²)𝑑𝜆,𝑡≥0,𝑘≥𝑘₀. One can simply prove that (𝑑^𝑗/𝑑𝑡^𝑗)𝑆^𝑘_𝜎(𝑡)=(1/2𝜋𝑖)∫_Γ𝑘,𝜎𝑒^𝜆𝑡𝜆^𝑗−2ℎ_𝜎(𝜆)𝑑𝜆,𝑡≥0,𝑘≥𝑘₀,𝑗∈ℕ. Let 𝑘₀<𝑘<𝑙. Then (iv) implies ‖𝑑^𝑗𝑑𝑡^𝑗𝑆^𝑘_𝜎𝑑(𝑡)−^𝑗𝑑𝑡^𝑗𝑆^𝑙_𝜎1(𝑡)‖=‖2𝜋_{Γ𝑙,𝜎∩{𝜆∈ℂ∶𝑘≤|𝜆|≤𝑙}}𝑒^𝜆𝑡𝜆^𝑗−2ℎ_𝜎≤𝑀(𝜆)𝑑𝜆‖_𝜎𝑒2𝜋^{2𝐶𝜎𝑡}_{Γ𝑙,𝜎∩{𝜆∈ℂ∶𝑘≤|𝜆|≤𝑙}}|Im𝜆|^1−𝜎𝑡|𝜆|^𝑗−2|𝑑𝜆|, (2.40) for all 𝑗∈ℕ₀. Since |Im𝜆|^1−𝜎𝑡|𝜆|^𝑗−2∼|Im𝜆|^{𝑗−1−𝜎𝑡},|𝜆|→∞,𝜆∈Γ_𝜎, one gets that, for every 𝑗∈ℕ₀ and 𝑡>𝑗/𝜎, the sequence ((𝑑^𝑗/𝑑𝑡^𝑗)𝑆^𝑘_𝜎(𝑡))_𝑘 is convergent in 𝐿(𝐸) and that the convergence is uniform on every compact subset of [𝑗/𝜎+𝜍,∞). Put 𝑆_𝑗,𝜎(𝑡)∶=lim_𝑘→∞(𝑑^𝑗/𝑑𝑡^𝑗)𝑆^𝑘_𝜎(𝑡),𝑗∈ℕ₀,𝑡>𝑗/𝜎. Then it is obvious that (𝑑/𝑑𝑡)𝑆_𝑗,𝜎(𝑡)=𝑆_{(𝑗+1),𝜎}(𝑡),𝑗∈ℕ₀,𝑡>(𝑗+1)/𝜎+𝜍. This implies that the mapping 𝑡↦𝑆_0,𝜎(𝑡),𝑡>(𝑗+1/𝜎)+𝜍 is 𝑗-times differentiable and that (𝑑^𝑗/𝑑𝑡^𝑗)𝑆_0,𝜎(𝑡)=𝑆_𝑗,𝜎(𝑡),𝑡>(𝑗+1)/𝜎)+𝜍. On the other hand, it is clear that, for every 𝜎>0,𝑥∈𝐷(𝐴²), and 𝜆∈{𝑧∈Ω_𝜎,𝜔∶𝑔_𝜎(𝑧)≠0},(𝐼−𝑔_𝜎(𝜆)𝐴)⁻¹𝐶𝑥=𝐶𝑥+𝑔_𝜎(𝜆)𝐶𝐴𝑥+𝑔_𝜎(𝜆)(𝐼−𝑔_𝜎(𝜆)𝐴)⁻¹𝐶𝐴²𝑥. (2.41) By (2.41), we get that, for every 𝑥∈𝐷(𝐴²) and 𝑡>0,𝑆_0,𝜎1(𝑡)𝑥=2𝜋𝑖_Γ𝜎𝑒^𝜆𝑡𝑓_𝜎(𝜆)𝐶𝑥+𝑔_𝜎(𝜆)𝐶𝐴𝑥+𝑔_𝜎(𝜆)(𝐼−𝑔_𝜎(𝜆)𝐴)⁻¹𝐶𝐴²𝑥𝜆²𝑑𝜆. (2.42) With (iv) and the residue theorem in view, it follows that, for every 𝑡>0 and 𝑥∈𝐷(𝐴²),𝑆_0,𝜎(𝑡)𝑥=^𝑡₀+1(𝑡−𝑠)𝑘(𝑠)𝑑𝑠𝐶𝑥+(𝑡∗𝑘∗𝑎)(𝑡)𝐶𝐴𝑥2𝜋𝑖^𝜔+𝑖∞_{𝜔−𝑖∞}𝑒^𝜆𝑡𝑓_𝜎(𝜆)𝑔_𝜎(𝜆)(𝐼−𝑔_𝜎(𝜆)𝐴)⁻¹𝐶𝐴²𝑥𝜆²𝑑𝜆. (2.43) Put 𝑅₂(𝑡)∶=∫^𝑡₀(𝑡−𝑠)𝑅(𝑠)𝑥𝑑𝑠,𝑥∈𝐸,𝑡≥0. By Theorem 2.6(i), we get that 𝑘^∼(𝜆)𝜆²(𝐼−𝑎^∼(𝜆)𝐴)⁻¹𝐶𝑥=^∞₀𝑒^−𝜆𝑡𝑅₂(𝑡)𝑥𝑑𝑡,𝑥∈𝐸,Re𝜆>𝜔,𝑘^∼(𝜆)≠0. (2.44) This implies that the function 𝜆↦ℎ_𝜎(𝜆)/𝜆² is bounded on some right half plane. Taking into account (2.41), we have that, for every 𝑡≥0 and 𝑥∈𝐷(𝐴²),𝑅₂(𝑡)=ℒ⁻¹(^∼𝑘(𝜆)𝜆²(𝐼−𝑎^∼(𝜆)𝐴)⁻¹𝐶𝑥)(𝑡)=^𝑡₀+1(𝑡−𝑠)𝑘(𝑠)𝑑𝑠𝐶𝑥+(𝑡∗𝑘∗𝑎)(𝑡)𝐶𝐴𝑥2𝜋𝑖^𝜔+𝑖∞_{𝜔−𝑖∞}𝑒^𝜆𝑡𝑓_𝜎(𝜆)𝑔_𝜎(𝜆)(𝐼−𝑔_𝜎(𝜆)𝐴)⁻¹𝐶𝐴²𝑥𝜆²𝑑𝜆. (2.45) By (2.43)-(2.45), 𝑆_0,𝜎(𝑡)=𝑅₂(𝑡),𝑡>0. The arbitrariness of 𝜎 implies that the mapping 𝑡↦𝑅₂(𝑡),𝑡>0 is infinitely differentiable, finishing the proof.

Using the argumentation given in [25], one can prove the following theorems.

Theorem 2.22. Suppose 𝑘(𝑡) and 𝑎(𝑡) satisfy (P1), 𝐴 is a subgenerator of a (local) (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)},𝜔≥max(0,abs(𝑘),abs(𝑎)), and 𝑚∈ℕ. Denote, for every 𝜀∈(0,1) and a corresponding 𝐾_𝜀>0,𝐹_𝜀,𝜔=∶{𝜆∈ℂ∶Re𝜆≥−ln𝜔_𝐿(𝐾_𝜀|Im𝜆|)+𝜔}. (2.46) Assume that, for every 𝜀∈(0,1), there exist 𝐶_𝜀>0,𝑀_𝜀>0, an open neighborhood 𝑂_𝜀,𝜔 of the region 𝐺_𝜀,𝜔∶={𝜆∈ℂ∶Re𝜆≥𝜔,𝑘^∼(𝜆)≠0}∪{𝜆∈𝐹_𝜀,𝜔∶Re𝜆≤𝜔}, and analytic mappings 𝑓_𝜀∶𝑂_𝜀,𝜔→ℂ,𝑔_𝜀∶𝑂_𝜀,𝜔→ℂ and ℎ_𝜀∶𝑂_𝜀,𝜔→𝐿(𝐸) such that (i)𝑓_𝜀(𝜆)=𝑘^∼(𝜆),Re𝜆>𝜔;𝑔_𝜀(𝜆)=𝑎^∼(𝜆),Re𝜆>𝜔,(ii)for every 𝜆∈𝐹_𝜀,𝜔, the operator 𝐼−𝑔_𝜀(𝜆)𝐴 is injective and Rang(𝐶)⊆Rang(𝐼−𝑔_𝜀(𝜆)𝐴),(iii)ℎ_𝜀(𝜆)=𝑓_𝜀(𝜆)(𝐼−𝑔_𝜀(𝜆)𝐴)⁻¹𝐶,𝜆∈𝐺_𝜀,𝜔,(iv)‖ℎ_𝜀(𝜆)‖≤𝑀_𝜀(1+|𝜆|)^𝑚𝑒^𝜀|Re𝜆|,𝜆∈𝐹_𝜀,𝜔,Re𝜆≤𝜔 and ‖ℎ_𝜀(𝜆)‖≤𝑀_𝜀(1+|𝜆|)^𝑚,𝜆∈ℂ,Re𝜆≥𝜔.Then (𝑅(𝑡))_{𝑡∈[0,𝜏)}is of class 𝐶^𝐿.

Theorem 2.23. Suppose 𝑘(𝑡) and 𝑎(𝑡) satisfy (P1), 𝐴 is a subgenerator of a (local) (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)},𝜔≥max(0,abs(𝑘),abs(𝑎)), and 𝑚∈ℕ. Denote, for every 𝜀∈(0,1),𝜌∈[1,∞) and a corresponding 𝐾_𝜀>0,𝐹_{𝜀,𝜔,𝜌}=∶{𝜆∈ℂ∶Re𝜆≥−𝐾_𝜀|𝐼𝑚𝜆|^1/𝜌+𝜔}. (2.47) Assume that, for every 𝜀∈(0,1), there exist 𝐶_𝜀>0,𝑀_𝜀>0, an open neighborhood 𝑂_𝜀,𝜔 of the region 𝐺_{𝜀,𝜔,𝜌}∶={𝜆∈ℂ∶Re𝜆≥𝜔,𝑘^∼(𝜆)≠0}∪{𝜆∈𝐹_{𝜀,𝜔,𝜌}∶Re𝜆≤𝜔}, and analytic mappings 𝑓_𝜀∶𝑂_𝜀,𝜔→ℂ,𝑔_𝜀∶𝑂_𝜀,𝜔→ℂ and ℎ_𝜀∶𝑂_𝜀,𝜔→𝐿(𝐸) such that the conditions (i)–(iv) of Theorem 2.22 hold with 𝐹_𝜀,𝜔, resp. 𝐺_𝜀,𝜔, replaced by 𝐹_{𝜀,𝜔,𝜌}, respectively, 𝐺_{𝜀,𝜔,𝜌}. Then (𝑅(𝑡))_{𝑡∈[0,𝜏)} is of class C^L.

Theorem 2.24. Suppose 𝛼>0,𝑗∈ℕ, and (𝑅(𝑡))_{𝑡∈[0,𝜏)} is a (local) (𝑎,𝑘)-regularized 𝐶-resolvent family with a subgenerator 𝐴. Set 𝑅_𝛼(𝑡)𝑥∶=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑅(𝑠)𝑥𝑑𝑠,𝑡∈[0,𝜏),𝑥∈𝐸. (2.48)
Then (𝑅(𝑡))_{𝑡∈[0,𝜏)} is an (𝑎,𝑘∗(𝑡^𝛼−1/Γ(𝛼)))-regularized 𝐶-resolvent family with a subgenerator 𝐴. Furthermore, if the mapping 𝑡↦𝑅(𝑡),𝑡∈(0,𝜏) is 𝑗-times differentiable, then the mapping 𝑡↦𝑅_𝛼(𝑡),𝑡∈(0,𝜏) is likewise 𝑗-times differentiable. If this isthe case, then we have, for every 𝑡∈[0,𝜏),𝑏∈(0,𝑡), and 𝑥∈𝐸∶
𝑑^𝑗𝑑𝑡^𝑗𝑅_𝛼(𝑡)𝑥=^𝑏₀(𝑡−𝑠)^{𝛼−1−𝑗}Γ(𝛼)^𝑗_𝑖=1(𝛼−𝑖)𝑅(𝑠)𝑥𝑑𝑠+^𝑗_𝑖=0(𝑡−𝑏)^{𝛼+𝑖−𝑗}×Γ(𝛼+𝑖+1)_{𝑗−1𝑘=0}(𝛼+𝑖−𝑘)𝑅^(𝑖)(𝑏)𝑥+^𝑡_𝑏(𝑡−𝑠)^𝛼𝑑Γ(𝛼+1)𝑑𝑠^𝑗𝑗𝑅(𝑠)𝑥𝑑𝑠, (2.49) and we have the following. (i)If (𝑅(𝑡))_{𝑡∈[0,𝜏)} is of class 𝐶^𝐿, resp. of class 𝐶_𝐿, then (𝑅_𝛼(𝑡))_{𝑡∈[0,𝜏)} is likewise of class 𝐶^𝐿, resptivley, of class 𝐶_𝐿.(ii)If (𝑅(𝑡))_{𝑡∈[0,𝜏)} is 𝜌-hypoanalytic, 1≤𝜌<∞, then (𝑅_𝛼(𝑡))_{𝑡∈[0,𝜏)} is likewise 𝜌-hypoanalytic.

Before going further, notice that we can slightly reformulate Theorem 2.16 and Theorems 2.21–2.23 in the case when the functions 𝑘^∼(𝜆) and 𝑎^∼(𝜆) possess the meromorphic extensions on the corresponding regions defined in formulation of mentioned theorems. Having in mind [9, Theorem ?2.1.1.11, Theorem ??2.1.1.14], we have the following interesting analogue of [25, Theorem ?2.8] which cannot be so easily interpreted in the case of general (𝑎,𝑘)-regularized 𝐶-resolvent families.

Theorem 2.25. (i) Let 𝐴 be a subgenerator of a local 𝐾-convoluted 𝐶-cosine function (𝐶_𝐾(𝑡))_{𝑡∈[0,𝜏)},0∈supp𝐾,𝐾∈𝐶^∞((0,𝜏)) (𝐾∈𝐶^𝑗((0,𝜏)),𝑗∈ℕ) resp. 𝐾 is of class 𝐶^𝐿 (𝐶_𝐿), and let 𝐾=𝐾_1|[0,𝜏) for an appropriate complex-valued function 𝐾₁∈𝐿¹_𝑙𝑜𝑐([0,2𝜏)). (Put Θ₁(𝑡)=∫^𝑡₀𝐾₁(𝑠)𝑑𝑠; since it makes no misunderstanding, we will also write 𝐾 and Θ, for 𝐾₁ and Θ₁, respectively, and denote by 𝐾∗𝐾 the restriction of this function to any subinterval of [0,2𝜏).) Let the mapping 𝑡↦𝐶_𝐾(𝑡),𝑡∈(0,𝜏), be infinitely differentiable (𝑗-times differentiable, 𝑗∈ℕ), respectively, and let (𝐶_𝐾(𝑡))_{𝑡∈[0,𝜏)} be of class 𝐶^𝐿 (𝐶_𝐿). Then 𝐴 is a subgenerator of a local (𝐾∗𝐾)-convoluted 𝐶²-cosine function (𝐶_𝐾∗𝐾(𝑡))_{𝑡∈[0,2𝜏)} satisfying that the mapping 𝑡↦𝐶_𝐾∗𝐾(𝑡),𝑡∈(0,2𝜏), is infinitely differentiable ((𝑗−1)-times differentiable), resp. (𝐶_𝐾∗𝐾(𝑡))_{𝑡∈[0,2𝜏)} is of class 𝐶^𝐿 (𝐶_𝐿). Furthermore, the suppositions 𝑗∈ℕ and 𝐾∈𝐶^𝑗((0,𝜏))∩𝐶^𝑗−1([0,𝜏)) imply the following: if the mapping 𝑡↦𝐶_𝐾(𝑡),𝑡∈(0,𝜏), is 𝑗-times differentiable, then the mapping 𝑡↦𝐶_𝐾∗𝐾(𝑡),𝑡∈(0,2𝜏) is likewise 𝑗-times differentiable.
(ii) Suppose 𝛼≥0, ?? 𝑗∈ℕ, and 𝐴 is a subgenerator of a local 𝛼-times integrated 𝐶-cosine function (𝐶_𝛼(𝑡))_{𝑡∈[0,𝜏)}. Then 𝐴 is a subgenerator of a local (2𝛼)-times integrated 𝐶²-cosine function (𝐶_2𝛼(𝑡))_{𝑡∈[0,2𝜏)} and the following holds.
(ii.1)If the mapping 𝑡↦𝐶_𝛼(𝑡),𝑡∈(0,𝜏) is infinitely differentiable (𝑗-times differentiable, 𝑗∈ℕ), then the mapping 𝑡↦𝐶_2𝛼(𝑡),𝑡∈(0,2𝜏) is infinitely differentiable ((𝑗−1)-times differentiable; 𝑗-times differentiable, provided 𝛼≥𝑗).(ii.2)If (𝐶_𝛼(𝑡))_{𝑡∈[0,𝜏)} is of class 𝐶^𝐿, resp. 𝐶_𝐿, then (𝐶_2𝛼(𝑡))_{𝑡∈[0,2𝜏)} is likewise of class 𝐶^𝐿, resp. 𝐶_𝐿.(ii.3)Assume 𝛼∈ℕ₀,𝑗∈ℕ, and the mapping 𝑡↦𝐶_𝛼(𝑡),𝑡∈(0,𝜏) is infinitely differentiable (𝑗-times differentiable). Then the mapping 𝑡↦𝐶_2𝛼(𝑡),𝑡∈(0,2𝜏), is 𝑗-times differentiable.

Proof. The first part of (i) can be proved by passing to the theory of semigroups (see [9, Theorem ?2.1.1.11] and [25, Theorem ?2.8]). So, let us assume 𝑗∈ℕ,𝐾∈𝐶^𝑗((0,𝜏))∩𝐶^𝑗−1([0,𝜏)),𝜏₀∈(0,𝜏), and let the mapping 𝑡↦𝐶_𝐾(𝑡),𝑡∈(0,𝜏) be 𝑗-times differentiable. By [9, Theorem ?2.1.1.14], 𝐴 is a subgenerator of a local (𝐾∗𝐾)-convoluted 𝐶²-cosine function (𝐶_𝐾∗𝐾(𝑡))_{𝑡∈[0,2𝜏)}, which is given by 𝐶_𝐾∗𝐾(𝑡)𝑥=^𝑡0𝐾(𝑡−𝑠)𝐶_𝐾(𝑠)𝐶𝑥𝑑𝑠,𝑡∈[0,𝜏₀],2𝐶_𝐾(𝜏₀)𝐶_𝐾(𝑡−𝜏₀)𝑥+(^𝑡−𝜏0₀+^𝜏0₀)𝐾(𝑡−𝑟)𝐶_𝐾−(𝑟)𝐶𝑥𝑑𝑟_{𝜏02𝜏0−𝑡}𝐾(𝑟+𝑡−2𝜏₀)𝐶_𝐾−(𝑟)𝐶𝑥𝑑𝑟_{𝑡−𝜏00}𝐾(𝑟+2𝜏₀−𝑡)𝐶_𝐾(𝑟)𝐶𝑥𝑑𝑟,𝑡∈(𝜏₀,2𝜏₀),𝑥∈𝐸. (2.50) Since the mapping 𝑡↦𝐶_𝐾(𝑡),𝑡∈(0,𝜏) is 𝑗-times differentiable and 𝐾∈𝐶^𝑗((0,∞)), we have that the mapping 𝑡↦𝐶_𝐾∗𝐾(𝑡),𝑡∈(0,𝜏) is also 𝑗-times differentiable. Arguing as in [25, Theorem ?2.8], one gets that the mappings 𝑡↦𝐶_𝐾(𝜏₀)𝐶_𝐾(𝑡−𝜏₀),𝑡∈(𝜏₀,2𝜏₀),𝑡↦(∫^𝑡−𝜏0₀+∫^𝜏0₀)𝐾(𝑡−𝑟)𝐶_𝐾(𝑟)𝐶𝑑𝑟,𝑡∈(𝜏₀,2𝜏₀) and 𝑡↦∫^𝑡−𝜏0₀𝐾(𝑟+2𝜏₀−𝑡)𝐶_𝐾(𝑟)𝐶𝑑𝑟,𝑡∈(𝜏₀,2𝜏₀) are 𝑗-times differentiable.
Let 𝑓(𝑡)∶=∫^𝜏0_{2𝜏0−𝑡}𝐾(𝑟+𝑡−2𝜏₀)𝐶_𝐾(𝑟)𝐶𝑑𝑟,𝑡∈(𝜏₀,2𝜏₀). Using the fact that 𝐾∈𝐶¹((0,𝜏))∩𝐶([0,𝜏)), we have 𝑓^′(𝑡)=∫^𝜏0_{2𝜏0−𝑡}𝐾^′(𝑟+𝑡−2𝜏₀)𝐶_𝐾(𝑟)𝐶𝑑𝑟+𝐾(0)𝐶_𝐾(2𝜏₀−𝑡)𝐶,𝑡∈(𝜏₀,2𝜏₀). Repeating this procedure leads us to the fact that the mapping 𝑡↦𝑓(𝑡),𝑡∈(𝜏₀,2𝜏₀) is 𝑗-times differentiable, and this completes the proof of (i). The proof of (ii) in the case 𝛼∈ℕ follows immediately from (i) with 𝐾(𝑡)=(𝑡^𝛼−1/Γ(𝛼)) while the proof of (ii) in the case 𝛼=0 is much easier [16].

Suppose that min(𝑎(𝑡),𝑘(𝑡))>0,𝑡∈(0,𝜏) and that 𝐴 is a subgenerator of an (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)}. We define the Favard class 𝐹_𝑎,𝑘 by setting 𝐹_𝑎,𝑘∶={𝑥∈𝐸∶sup_{𝑡∈(0,𝜏)}‖𝑅(𝑡)𝑥−𝑘(𝑡)𝐶𝑥‖(𝑎∗𝑘)(𝑡)<∞}. (2.51) Equipped with the norm |⋅|_𝑎,𝑘∶=‖⋅‖+sup_{𝑡∈(0,𝜏)}(‖𝑅(𝑡)⋅−𝑘(𝑡)𝐶⋅‖/(𝑎∗𝑘)(𝑡)),𝐹_𝑎,𝑘 becomes a Banach space, and in the case when ‖𝑅(𝑡)‖=𝑂(𝑘(𝑡)),𝑡∈[0,𝜏), we have 𝐷(𝐴)⊆𝐹_𝑎,𝑘. The proof of [5, Theorem ?3.4] immediately implies the following assertion.

Theorem 2.26. Assume min(𝑎(𝑡),𝑘(𝑡))>0,𝑡∈(0,𝜏),abs(𝑘)=abs(𝑎)=0,𝐴 is a subgenerator of an (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_𝑡≥0 satisfying ‖𝑅(𝑡)‖=𝑂(1),𝑡≥0 and (H5) holds. (i)Let 𝑥∈𝐹_𝑎,𝑘. Then sup_{𝜆>0,∼𝑘(𝜆)≠0}‖𝐴(𝐼−𝑎^∼(𝜆)𝐴)⁻¹𝐶𝑥‖<∞. (2.52)(ii)Assume, in addition, that the mapping 𝑎^∼∶(0,∞)→(0,∞) is surjective and that sup_t>0(1∗𝑎)(𝑡)/(𝑎∗𝑘)(𝑡)<∞. Then (2.52) implies 𝐶𝑥∈𝐹_𝑎,𝑘.

The assertion (ii) of the next theorem improves [5, Theorem ?4.6].

Theorem 2.27 (cf. [26, Theorem ?4.2] and Proposition? 2.12.7). (i) Suppose 𝐴 is a subgenerator of a (local, global exponentially bounded) (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)},𝐷(𝐴) and Rang(𝐶) are dense in 𝐸 and 𝛼>0. Then 𝐴^∗ is a subgenerator of a (local, global exponentially bounded) (𝑎,𝑘∗₀(𝑡^𝛼−1/Γ(𝛼)))-regularized 𝐶^∗-resolvent family (𝑅^∗_𝛼(𝑡))_{𝑡∈[0,𝜏)}, which is given by 𝑅_𝛼(𝑡)𝑥^∗∶=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑅(𝑠)^∗𝑥^∗𝑑𝑠,𝑡∈[0,𝜏),𝑥^∗∈𝐸^∗. (2.53)
(ii) Suppose 𝐴 is a subgenerator of a (local, global exponentially bounded) (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)}, and 𝐷(𝐴) and Rang(𝐶) are dense in 𝐸. Then the part of 𝐴^∗ in 𝐷(𝐴^∗) is a subgenerator of a (local, global exponentially bounded) (𝑎,𝑘)-regularized 𝐶^∗_{𝐷(𝐴∗)}-resolvent family in E^∗.
(iii) Suppose 𝐸 is reflexive, 𝐷(𝐴) and Rang(𝐶) are dense in 𝐸,𝑘(𝑡) and 𝑎(𝑡) satisfy (P1), and 𝐴 is a subgenerator of a (local, global exponentially bounded) (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)}. Then 𝐴^∗ is a subgenerator of a (local, global exponentially bounded) (𝑎,𝑘)- regularized 𝐶^∗-resolvent family (of the same exponential type, in the second case).

Suppose, for the time being, that 𝑎∈𝐶([0,𝜏)) and denote, for every 𝜆∈ℂ, by 𝑠(𝑡,𝜆) the unique continuous solution of the equation 𝑠(𝑡,𝜆)=𝑎(𝑡)+𝜆^𝑡₀𝑎(𝑡−𝑣)𝑠(𝑣,𝜆)𝑑𝑣,𝑡∈[0,𝜏). (2.54) Put 𝑟(𝑡,𝜆)∶=𝑘(𝑡)+𝜆∫^𝑡₀𝑠(𝑡−𝑣,𝜆)𝑘(𝑣)𝑑𝑣,𝑡∈[0,𝜏). Arguing as in [5, Section ?5], one can simply verify the validity of the next theorem.

Theorem 2.28. (i) Let 𝐴 be a subgenerator of an (𝑎,𝑘)- regularized 𝐶-resolvent family(𝑅(𝑡))_{𝑡∈[0,𝜏)}, and let (H5) hold. If the operator 𝑟(𝑡,𝜆)𝐶−𝑅(𝑡) is bijective for some 𝑡∈[0,𝜏) and 𝜆∈ℂ, then 𝜆∈𝜌(𝐴).
(ii) Let 𝐴 be a densely defined subgenerator of an (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)}, and let (H5) hold. If Rang(𝜆−𝐴)≠𝐸 for some 𝜆∈ℂ, then, for every 𝑡∈[0,𝜏),Rang(𝑟(𝑡,𝜆)𝐶−𝑅(𝑡))≠𝐸.
(iii) Let 𝐴 be a subgenerator of an (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)}. Then the assumption 𝐴𝑥=𝜆𝑥, for some 𝑥∈𝐸 and 𝜆∈ℂ, implies 𝑟(𝑡,𝜆)𝐶𝑥=𝑅(𝑡)𝑥,𝑡∈[0,𝜏).

For further information concerning duality and spectral properties of (𝑎,𝑘)-regularized resolvent families, we refer to [5].

Proposition 2.29 (cf. [9, Proposition ?2.1.1.17]). Suppose ±𝐴 are subgenerators of (local, global exponentially bounded) (𝑎,𝑘)- regularized 𝐶- resolvent families (𝑅^±(𝑡))_{𝑡∈[0,𝜏)}, and 𝐴² is closed. Then 𝐴² is a subgenerator of a (local, global exponentially bounded) (𝑎∗𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)}, which is given by 𝑅(𝑡)𝑥∶=1/2(𝑅⁺(𝑡)𝑥+𝑅⁻(𝑡)𝑥),𝑥∈𝐸,𝑡∈[0,𝜏).

Proof. Clearly, (𝑅(𝑡))_{𝑡∈[0,𝜏)} is a strongly continuous operator family, 𝐶𝐴²⊆𝐴²𝐶,𝑅(0)=𝑘(0)𝐶,𝑅(𝑡)𝐶=𝐶𝑅(𝑡) and 𝑅(𝑡)𝐴²⊆𝐴²𝑅(𝑡),𝑡∈[0,𝜏). Let 𝑥∈𝐷(𝐴²). Then we have (𝑎∗𝑎∗𝑅)(𝑡)𝐴²1𝑥=2(𝑎∗𝑎∗𝑅⁺1𝐴)(𝑡)𝐴𝑥+2(𝑎∗𝑎∗𝑅⁻=1𝐴)(𝑡)𝐴𝑥2(𝑎∗(𝑅⁺1−𝑘𝐶))(𝑡)𝐴𝑥+2(𝑎∗(𝑅⁻=1−𝑘𝐶))(𝑡)𝐴𝑥2(𝑎∗𝑅⁺1)(𝑡)𝐴𝑥+2(𝑎∗𝑅⁻=1)(𝑡)𝐴𝑥2(𝑅⁺1(𝑡)𝑥−𝑘(𝑡)𝐶𝑥)+2(𝑅⁻(𝑡)𝑥−𝑘(𝑡)𝐶𝑥)=𝑅(𝑡)𝑥−𝑘(𝑡)𝐶𝑥,𝑡∈[0,𝜏). (2.55) This completes the proof.

The next version of the abstract Weierstrass formula extends [15, Theorem ?11].

Theorem 2.30. (i) Assume that 𝑘(𝑡) and 𝑎(𝑡) satisfy (P1), and there exist 𝑀>0 and 𝜔>0 such that |𝑘(𝑡)|≤𝑀𝑒^𝜔𝑡,𝑡≥0. Assume, further, that there exist a number 𝜔^′≥𝜔 and a function 𝑎₁(𝑡) satisfying (P1) and 𝑎^∼₁(𝜆)=𝑎^∼(√𝜆),𝜆∈ℂ,Re𝜆>𝜔^′. Let 𝐴 be a subgenerator of an exponentially bounded (𝑎,𝑘)-regularized 𝐶-resolvent family (𝐶(𝑡))_𝑡≥0, and let (H5) hold. Then 𝐴 is a subgenerator of an exponentially bounded, analytic (𝑎₁,𝑘₁)-regularized 𝐶-resolvent family (𝑅(𝑡))_𝑡≥0 of angle (𝜋/2), where
𝑘₁(𝑡)∶=^∞₀𝑒^{−𝑠2/4𝑡}√𝜋𝑡𝑘(𝑠)𝑑𝑠,𝑡>0,𝑘₁(0)∶=𝑘(0),𝑅(𝑡)𝑥∶=^∞₀𝑒^{−𝑠2/4𝑡}√𝜋𝑡𝐶(𝑠)𝑥𝑑𝑠,𝑡>0,𝑥∈𝐸,𝑅(0)∶=𝑘(0)𝐶. (2.56)(2.57)
(ii) Assume 𝑘(𝑡) satisfy (P1), 𝛽>0, and there exist 𝑀>0 and 𝜔>0 such that |𝑘(𝑡)|≤𝑀𝑒^𝜔𝑡,𝑡≥0. Let 𝐴 be a subgenerator of an exponentially bounded (𝑡^2𝛽−1/Γ(2𝛽),𝑘)-regularized 𝐶-resolvent family (𝐶(𝑡))_𝑡≥0, and let (H5) hold. Then 𝐴 is a subgenerator of an exponentially bounded, analytic (𝑡^𝛽−1/Γ(𝛽),𝑘₁)-regularized 𝐶-resolvent family (𝑅(𝑡))_𝑡≥0 of angle 𝜋/2, where 𝑘₁(𝑡) and 𝑅(𝑡) are defined through (2.56) and (2.57).

Proof. Since 𝑘(𝑡) is continuous and exponentially bounded, one can use the substitution 𝑟=𝑠/√𝑡 and the dominated convergence theorem after that to deduce that, for every 𝑠≥0,𝑘₁(𝑡)=^∞₀𝑒^−𝑟2/4√𝜋𝑘(𝑟√𝑡)𝑑𝑟→𝑘₁(𝑠),𝑡→𝑠. (2.58) This implies 𝑘₁∈𝐶([0,∞)). Moreover, 𝑘₁(𝑡) is a kernel since limsup_𝜆→+∞ln|𝑘^∼₁(𝜆)|𝜆=limsup_𝜆→+∞ln|𝑘^∼(√𝜆)/√𝜆|𝜆=0. (2.59) Let 𝑥∈𝐸 be fixed. Then, for every 𝑠≥0,𝑅(𝑡)𝑥=^∞₀𝑒^−𝑟2/4√𝜋𝐶(𝑟√𝑡)𝑥𝑑𝑟→𝑅(𝑠)𝑥,𝑡→𝑠. (2.60) By (2.60), (𝑅(𝑡))_𝑡≥0 is a strongly continuous, exponentially bounded operator family. Furthermore, one can employ Theorem 2.6(i) and [12, Proposition ??1.6.8] to obtain that, for every 𝜆∈ℂ with Re𝜆>𝛽² and 𝑘^∼₁(𝜆)≠0,^∞₀𝑒^−𝜆𝑡𝑅(𝑡)𝑥𝑑𝑡=^∞₀𝑒^−𝜆𝑡1√𝜋𝑡^∞0𝑒^{−𝑠2/4𝑡}=1𝐶(𝑠)𝑥𝑑𝑠𝑑𝑡√𝜆^∞0𝑒^{−√𝜆𝑠}=1𝐶(𝑠)𝑥𝑑𝑠√𝜆^∼𝑘(√𝜆)(𝐼−𝑎^∼(√𝜆)𝐴)⁻¹=𝐶𝑥_∼𝑘₁(𝜆)(𝐼−𝑎^∼₁(𝜆)𝐴)⁻¹𝐶𝑥. (2.61) By Theorem 2.6(ii), we get that (𝑅(𝑡))_𝑡≥0 is an exponentially bounded (𝑎₁,𝑘₁)-regularized 𝐶-resolvent family with a subgenerator 𝐴, and the remnant of the proof of (i) may be carried out by modifying the corresponding part of the proof of [15, Theorem ?11]; the assertion (ii) follows from (i) with 𝑎(𝑡)=𝑡^2𝛽−1/Γ(2𝛽).

Notice that 𝑎₁(𝑡)=∫^∞₀𝑠(𝑒^{−𝑠2/4𝑡}/2√𝜋𝑡^3/2)𝑎(𝑠)𝑑𝑠,𝑡>0, whenever the function 𝑎(𝑡) is exponentially bounded.

Example 2.31. (i)(Reference[15]) Let 𝐸=𝐿^𝑝(ℝ),1≤𝑝≤∞,𝛼∈(−1,1), and 𝑎(𝑡)=(𝑡^𝛼/Γ(𝛼+1)). Consider the next multiplication operator with maximal domain in 𝐸:
𝐴𝑓(𝑥)∶=(1+𝑥+𝑖𝑥²)^𝛼+1𝑓(𝑥),𝑥∈ℝ,𝑓∈𝐸. (2.62) Assume 𝑠∈(1,2),𝛿=1/𝑠,𝑀_𝑝=𝑝!^𝑠, and 𝐾_𝛿(𝑡)∶=ℒ⁻¹(𝑒^−𝜆𝛿)(𝑡),𝑡≥0. Then 𝐴 generates a global (not exponentially bounded) (𝑎,𝐾_𝛿)-regularized resolvent family since, for every 𝜏∈(0,∞),𝐴 generates a local (𝑎,𝐾_𝛿)-regularized resolvent family on [0,𝜏). In order to show this, designate by 𝑀(𝑡) the associated function of the sequence (𝑀_𝑝) and put Λ_{𝛼,𝛽,𝛾}∶={𝜆∈ℂ∶Re𝜆≥(𝑀(𝛼𝜆)/𝛾)+𝛽},𝛼>0,𝛽>0,𝛾>0. Clearly, there exists a constant 𝐶_𝑠>0 such that 𝑀(𝜆)≤𝐶_𝑠|𝜆|^1/𝑠,𝜆∈ℂ. Given 𝜏>0, choose 𝛼>0 and 𝛽>0 such that 𝜏≤cos(𝛿𝜋/2)/𝐶_𝑠𝛼^1/𝑠 as well as that Λ_𝛼,𝛽,1⊆𝜌(𝐴) and that the resolvent of 𝐴 is bounded on the set {𝜆^𝛼+1∶𝜆∈Λ_𝛼,𝛽,1}. Put Γ∶=𝜕(Λ_𝛼,𝛽,1), and assume that the curve Γ is upward oriented. Define, for every 𝑓∈𝐸,𝑥∈ℝ and 𝑡∈[0,cos(𝛿𝜋/2)/𝐶_𝑠𝛼^1/𝑠),(𝑅_𝛿1(𝑡)𝑓)(𝑥)∶=2𝜋𝑖_Γ𝜆^𝛼+1𝑒^{𝜆𝑡−𝜆𝛿}𝜆^𝛼+1−(1+𝑥+𝑖𝑥²)^𝛼+1𝑑𝜆𝑓(𝑥). (2.63) Then one can straightforwardly check that (𝑅_𝛿(𝑡))_{𝑡∈[0,𝜏)} is a local (𝑎,𝐾_𝛿)-regularized resolvent family generated by 𝐴. Arguing in the same way, we get that there exists 𝜏₀>0 such that 𝐴 generates a local (𝑎,𝐾_1/2)-regularized resolvent family on [0,𝜏₀), where 𝐾_1/2(𝑡)∶=ℒ⁻¹(𝑒^−𝜆1/2)(𝑡),𝑡≥0.
(ii) [References [9, 27]] Let 𝐴_(𝑝!^𝑠₎ and 𝐸_(𝑝!^𝑠₎ be as in [27, Example ?1.6] with 𝑀_𝑝=𝑝!^𝑠(𝑠>1). Let 𝛽∈(0,1), and let, for every 𝑙>0,𝑘_𝑙(𝑡)=ℒ⁻¹(1/∏^∞_𝑝=1(1+𝑙𝜆/𝑝^𝑠/𝛽))(𝑡),𝑡≥0, (see (2.36)-(2.37) and 𝑎(𝑡)=𝑡^𝛽−1/Γ(𝛽). Then it is obvious that there exist 𝑙^′>0 and 𝐾>0 such that ‖𝜆𝑘^∼_𝑙^′(𝜆)(𝐼−𝑎^∼(𝜆)𝐴)⁻¹‖≤𝐾,𝜆∈Σ_𝜋/2𝛽. This in combination with Theorem 2.17 implies that, for every 𝑙>𝑙^′, the operator 𝐴_(𝑝!^𝑠₎ generates an analytic (𝑎,𝑘_𝑙)-regularized resolvent of angle (𝜋/2)((1/𝛽)−1). In the meantime, 𝐴_(𝑝!^𝑠₎ does not generate an exponentially bounded (𝑎,𝑡^𝛼/Γ(𝛼+1))-regularized resolvent (𝛼≥0) since 𝐴_(𝑝!^𝑠₎ is not stationary dense.
(iii) [References [9, 28]; cf. also [29, Example ?2.20]] Suppose 𝐸∶=𝐿²[0,𝜋] and 𝐴∶=−Δ with the Dirichlet or Neumann boundary conditions, 𝛽∈[1/2,1),𝛼>1+𝛽,𝑎(𝑡)=(𝑡^𝛽−1/Γ(𝛽)), and ℎ_𝛼,𝛽1(𝜆)∶=𝜆^𝛼∞_𝑛=0𝑛²−𝜆^𝛽𝑛²+𝜆^𝛽,Re𝜆>0,𝜆≠𝑛^2/𝛽,𝑛∈ℕ. (2.64)
Define 𝐡_𝛼,𝛽∶Σ_𝜋/2𝛽→ℂ by setting: 𝐡_𝛼,𝛽(𝜆)=ℎ_𝛼,𝛽(𝜆),𝜆∈Σ_𝜋/2𝛽,𝜆≠𝑛^2/𝛽,𝑛∈ℕ, and 𝐡_𝛼,𝛽(𝑛^2/𝛽)=0,𝑛∈ℕ. Then the function 𝐡_𝛼,𝛽(𝜆) is analytic, and there exists a constant 𝐶>0 such that ‖𝐡_𝛼,𝛽(𝜆)(𝐼−𝑎^∼(𝜆)𝐴)⁻¹‖≤1+𝐶|𝜆|^𝛽|𝜆|^𝛼,𝜆∈Σ_𝜋/2𝛽. (2.65) Let 𝑘(𝑡)=ℒ⁻¹(𝐡_𝛼,𝛽(𝜆))(𝑡),𝑡≥0. By Theorem 2.17, it follows that 𝐴 generates an exponentially bounded, analytic (𝑎,𝑘)-regularized resolvent (𝑅(𝑡))_𝑡≥0 of angle (𝜋/2)(1/𝛽−1). Using the inverse Laplace transform, one can simply prove that ‖𝑅(𝑡)‖=𝑂(𝑡^𝛼−1+𝑡^{𝛼+𝛽−1}),𝑡≥0. Since Δ generates a cosine function, we are in a position to apply Theorem 2.17 to deduce that Δ generates an exponentially bounded, analytic (𝑎,𝑘)-regularized resolvent of angle (𝜋/2)(1/𝛽−1). By Proposition 2.29, we have that the biharmonic operator Δ², equipped with the suitable boundary conditions, generates an exponentially bounded, analytic (𝑎∗𝑎,𝑘)-regularized resolvent of angle (𝜋/2)(1/𝛽−1). Then the use of Theorem 2.30(ii) enables one to see that there exists a continuous kernel 𝑘₁(𝑡) such that Δ² generates an exponentially bounded, analytic (𝑎,𝑘₁)-regularized resolvent family of angle 𝜋/2. Keeping in mind the fact that −Δ^2𝑛 generates an analytic 𝐶₀-semigroup of angle 𝜋/2 (cf. for example [30, page 215]), one can prove that, for every 𝑛∈ℕ, there exists an exponentially bounded kernel 𝑘_𝑛(𝑡) such that the polyharmonic operator Δ^2𝑛 generates an exponentially bounded, (𝑎,𝑘_𝑛)-regularized resolvent family of angle 𝜋/2 [15].
It has recently been proved that, in the case 𝛽=1, there exists an exponentially bounded continuous kernel 𝐾(𝑡) such that 𝐴 generates an exponentially bounded, analytic 𝐾-convoluted semigroup of angle 𝜋/2 [25]. Let us consider now the case 𝛽∈(1,2) and 𝑎(𝑡)=(𝑡^𝛽−1/Γ(𝛽)). Choose a number 𝑎∈(1/2,1/𝛽) and after that a number 𝑠∈(1,1/𝛽𝑎). Put 𝑘_𝑙(𝑡)∶=ℒ⁻¹(1/∏^∞_𝑝=11+𝑙𝜆/𝑝^𝑠)(𝑡),𝑡≥0. Arguing as in [25], one yields that the function 𝐡(𝜆)=∏^∞_𝑛=0(𝑛²−𝜆)/(𝑛²+𝜆),𝜆∈ℂ⧵{±𝑛²∶𝑛∈ℕ₀};𝐡(𝑛²)=0,𝑛∈ℕ, is analytic, and that there exists 𝑀>0 such that, for every 𝛾∈(0,𝜋(1/𝛽−1/2)), there exist 𝑀_𝛾>0 and 𝑐_𝛾>0 such that, for every 𝜆∈Σ_𝛾,
‖𝐡(𝜆^𝑏)(𝐼−𝑎^∼(𝜆)𝐴)⁻¹‖≤|𝜆|^𝑏1max((𝑀+|𝜆|^𝛽),𝑀_𝛾(⌈2|𝜆|^𝛽+1⌉|𝜆|^𝛽+𝜋²3)exp(𝑐_𝛾|𝜆|^𝛽𝑎)). (2.66) Furthermore, there exists an exponentially bounded continuous kernel 𝑘(𝑡) such that 𝑘^∼(𝜆)=𝑘^∼_𝑙(𝜆)𝐡(𝜆^𝛽),𝜆∈ℂ,Re𝜆>0. By (2.66), it follows that 𝐴 generates an exponentially bounded, analytic (𝑎,𝑘)-regularized resolvent of angle 𝜋(1/𝛽−1/2). Furthermore, an application of Theorem 2.17 gives that Δ generates an exponentially bounded, analytic (𝑎,𝑘)-regularized resolvent of angle 𝜋(1/𝛽−1/2). By Proposition 2.29, we have that Δ² generates an exponentially bounded, analytic (𝑎∗𝑎,𝑘)-regularized resolvent of angle 𝜋(1/𝛽−1/2). Arguing as in the case 𝛽∈[1/2,1), we have that, for every 𝑛∈ℕ, there exists an exponentially bounded kernel 𝑘_𝑛(𝑡) such that the polyharmonic operator Δ^2𝑛 generates an exponentially bounded, (𝑎,𝑘_𝑛)-regularized resolvent family of angle 𝜋/2. In the case 𝛽=2, it is known that 𝐴 cannot be the generator of any exponentially bounded convoluted cosine function [15]; the case 𝛽∈(0,1/2) requires an additional analysis. Finally, it is worth noting that we can incorporate the above results in the study of the equation 𝐃^𝛽_𝑡𝑢(𝑡,𝑥)=(−Δ)^2𝑛𝑢(𝑡,𝑥),𝑥∈(0,𝜋),𝑡>0(𝑛∈ℕ₀), (2.67) where 𝐃^𝛽_𝑡 denotes the Caputo fractional derivative [29].
The next theorem generalizes [6, Theorem ?3.6, Corollary ??3.8] (cf. also [31, Theorem ?2.1] and [32, Theorem ?3]).

Theorem 2.32. (i) Assume 𝐶([0,∞))∋𝑎 satisfies (P1), (H5) holds, 𝐵∈𝐿(𝐸),Rang(𝐵)⊆Rang(𝐶) that and 𝐴 is a subgenerator of an exponentially bounded (𝑎,𝑎)-regularized 𝐶-resolvent family (𝑅(𝑡))_𝑡≥0. Assume, further, that there exists 𝜔≥0 such that, for every ℎ≥0 and for every function 𝑓∈𝐶([0,∞)∶𝐸),
(Ma)∫^ℎ₀𝑅(ℎ−𝑠)𝐶⁻¹𝐵𝑓(𝑠)𝑑𝑠∈𝐷(𝐴),(Mb)‖𝐴∫^ℎ₀𝑅(ℎ−𝑠)𝐶⁻¹𝐵𝑓(𝑠)𝑑𝑠‖≤𝑒^𝜔𝑡𝜇_𝐵(ℎ)‖𝑓‖_[0,ℎ],𝑡≥0, where ‖𝑓‖_[0,ℎ]∶=sup_{𝑡∈[0,ℎ]}‖𝑓(𝑡)‖,𝜇_𝐵(𝑡)∶[0,∞)→[0,∞) is continuous, nondecreasing and satisfies 𝜇_𝐵(0)=0,(Mc)there exists an injective operator 𝐶₁∈𝐿(𝐸) such that Rang(𝐶₁)⊆Rang(𝐶) and that 𝐶₁𝐴(𝐼+𝐵)⊆𝐴(𝐼+𝐵)𝐶₁.Then 𝐴(𝐼+𝐵) is a subgenerator of an exponentially bounded (𝑎,𝑎)-regularized 𝐶₁-resolvent family (𝑆(𝑡))_𝑡≥0 which satisfies the following integral equation 𝑆(𝑡)𝑥=𝑅(𝑡)𝐶⁻¹𝐶₁𝑥+𝐴^𝑡₀𝑅(𝑡−𝑠)𝐶⁻¹𝐵𝑆(𝑠)𝑥𝑑𝑠,𝑡≥0,𝑥∈𝐸. (2.68)
(ii) Let 𝐴 be a subgenerator of an exponentially bounded, once integrated 𝐶-cosine function and let 𝜔,𝐵, and 𝐶₁ be as in (i). Then 𝐴(𝐼+𝐵) is a subgenerator of an exponentially bounded, once integrated 𝐶₁-cosine function.

Remark 2.33. (i) Assume that 𝐴 is a subgenerator of an exponentially bounded (𝑎,𝑎)-regularized 𝐶-resolvent family (𝑅(𝑡))_𝑡≥0 and that a Banach space (𝑍,|⋅|_𝑍) satisfies the conditions (Za), (Zb), and (Zc) given in the formulation of [6, Definition ?4.1]. (In particular, these conditions hold for [𝐷(𝐴)].) Then (Ma) and (Mb) are fulfilled if 𝐶⁻¹𝐵∈𝐿(𝑋,𝑍).
(ii) (References [32, 33]) Let 𝐵∈𝐿(𝐸), and let 𝐵𝐶=𝐶𝐵.
(ii.1)Assume that 𝐵𝐴 is a subgenerator of a (local) (𝑎,𝑘)-regularized 𝐶-resolvent family, and (H5) holds for 𝐵𝐴 and 𝐶. Then 𝐴𝐵 is a subgenerator of an (𝑎,𝑘)-regularized 𝐶-resolvent family.(ii.2)Assume that 𝐴𝐵 is a subgenerator of a (local) (𝑎,𝑘)-regularized 𝐶-resolvent family and (H5) holds for 𝐴𝐵 and 𝐶. Then 𝐵𝐴 is a subgenerator of an (𝑎,𝑘)-regularized 𝐶-resolvent family, provided 𝜌(𝐵𝐴)≠∅.

The proof of the next generalization of [15, Proposition ?3] is provided for the sake of completeness.

Theorem 2.34. Assume that 𝜏∈(0,∞],𝐿¹_loc([0,𝜏))∋𝑎₁ is a kernel, 𝐿¹_loc([0,𝜏))∋𝑘??is a kernel, 𝑎(𝑡)=(𝑎₁∗𝑎₁)(𝑡),𝑡∈[0,𝜏), and 𝑘₁(𝑡)=(𝑘∗𝑎₁)(𝑡),𝑡∈[0,𝜏). Put 𝒜≡(^0𝐼_𝐴0),𝒞≡(^𝐶0_0𝐶), and assume that (H5) holds. Then 𝐴 is a subgenerator of an (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)} if and only if 𝒜 is a subgenerator of an (𝑎₁,𝑘₁)-regularized 𝒞-resolvent family (𝑆(𝑡))_{𝑡∈[0,𝜏)}. If this is the case, then we have 𝑆(𝑡)=(𝑎₁∗𝑅)(𝑡)(𝑎∗𝑅)(𝑡)𝑅(𝑡)−𝑘(𝑡)𝐶(𝑎₁∗𝑅)(𝑡),0≤𝑡<𝜏, (2.69) and the integral generators of (𝑅(𝑡))_{𝑡∈[0,𝜏)} and (𝑆(𝑡))_{𝑡∈[0,𝜏)}, denoted respectively by 𝐵 and ℬ, satisfy ℬ=(^0𝐼_𝐵0).

Proof. It is immediately verified that (𝑆(𝑡))_{𝑡∈[0,𝜏)} is a nondegenerate, strongly continuous operator family in 𝐸×𝐸 which satisfies 𝑆(𝑡)𝒜⊆𝒜𝑆(𝑡) and 𝑆(𝑡)𝒞=𝒞𝑆(𝑡),0≤𝑡<𝜏. Furthermore, the function 𝑘₁(𝑡) is a continuous kernel, 𝑆(0)=0=𝑘₁(0)𝒞, and 𝒞𝒜⊆𝒜𝒞. Let 𝑥∈𝐷(𝐴), and let 𝑦∈𝐸. Then a simple computation involving (H5) shows that, for every 𝑡∈[0,𝜏),^𝑡₀𝑎₁𝑥𝑦(𝑡−𝑠)𝒜𝑆(𝑠)𝑑𝑠=^𝑡₀𝑎₁(𝑡−𝑠)𝑅(𝑠)𝑥−𝑘(𝑠)𝐶𝑥+(𝑎₁∗𝑅)(𝑠)𝑦𝐴(𝑎₁=∗𝑅)(𝑠)𝑥+𝐴(𝑎∗𝑅)(𝑠)𝑦𝑑𝑠(𝑎₁∗𝑅)(𝑡)𝑥−(𝑎₁∗𝑘)(𝑡)𝐶𝑥+(𝑎₁∗𝑎₁∗𝑅)(𝑡)𝑦𝐴(𝑎∗𝑅)(𝑡)𝑥+(𝑎₁=∗(𝑅−𝑘𝐶))(𝑡)𝑦(𝑎₁∗𝑅)(𝑡)𝑥−(𝑎₁∗𝑘)(𝑡)𝐶𝑥+(𝑎₁∗𝑎₁∗𝑅)(𝑡)𝑦𝑅(𝑡)𝑥−𝑘(𝑡)𝐶𝑥+(𝑎₁𝑥𝑦∗(𝑅−𝑘𝐶))(𝑡)𝑦=𝑆(𝑡)−𝑘₁𝑥𝑦(𝑡)𝒞. (2.70) Assume now that 𝒜 is a subgenerator of an (𝑎₁,𝑘₁)-regularized 𝒞-resolvent family (𝑆(𝑡))_{𝑡∈[0,𝜏)}. Put 𝑆(𝑡)=(^{𝑆1(𝑡)𝑆2}_𝑆^(𝑡)_{3(𝑡)𝑆4(𝑡)})_{𝑡∈[0,𝜏)}, where 𝑆^𝑖(𝑡)∈𝐿(𝐸),𝑖∈{1,2,3,4}, and 0≤𝑡<𝜏. A simple consequence of 𝑆(𝑡)𝒞=𝒞𝑆(𝑡),𝑡∈[0,𝜏) is 𝑆^𝑖(𝑡)𝐶=𝐶𝑆^𝑖(𝑡),𝑡∈[0,𝜏),𝑖∈{1,2,3,4}. Since 𝑆(𝑡)𝒜⊆𝒜𝑆(𝑡),𝑡∈[0,𝜏), one gets 𝑆¹(𝑡)𝑥+𝑆²𝑆(𝑡)𝑦∈𝐷(𝐴),₁(𝑡)𝑦+𝑆²(𝑡)𝐴𝑥=𝑆³(𝑡)𝑥+𝑆⁴𝑆(𝑡)𝑦,₃(𝑡)𝑦+𝑆⁴(𝑡)𝐴𝑥=𝐴(𝑆¹(𝑡)𝑥+𝑆²(𝑡)𝑦),0≤𝑡<𝜏,𝑥∈𝐷(𝐴),𝑦∈𝐸. (2.71) Hence, 𝑆³(𝑡)𝑥=𝑆²(𝑡)𝐴𝑥,𝑥∈𝐷(𝐴), and 𝑆³(𝑡)𝑦=𝐴𝑆²(𝑡)𝑦,𝑦∈𝐸,0≤𝑡<𝜏. This implies that, for every 𝑥∈𝐷(𝐴),𝑆³(𝑡)𝐴𝑥=𝐴𝑆²(𝑡)𝐴𝑥=𝐴𝑆³(𝑡)𝑥,𝑡∈[0,𝜏). Thereby, 𝑆³(𝑡)𝐴⊆𝐴𝑆³(𝑡),𝑡∈[0,𝜏), and (𝑅(𝑡)≡𝑆³(𝑡)+𝑘(𝑡)𝐶)_{𝑡∈[0,𝜏)} is a strongly continuous operator family in 𝐸 satisfying 𝑅(0)=𝑘(0)𝐶,𝑅(𝑡)𝐶=𝐶𝑅(𝑡) and 𝑅(𝑡)𝐴⊆𝐴𝑅(𝑡),0≤𝑡<𝜏. Since, for every 𝜆∈ℂ,𝜆∈𝜌_𝒞(𝒜) if and only if 𝜆²∈𝜌_𝐶(𝐴) [25], we have that (H5) holds for 𝒜 and 𝒞. Since, for every 𝑥∈𝐸 and 𝑦∈𝐸,𝒜(𝑎₁∗𝑆₁)(𝑡)(𝑎₁∗𝑆₂)(𝑡)(𝑎₁∗𝑆₃)(𝑡)(𝑎₁∗𝑆₄𝑥𝑦=𝑆)(𝑡)¹(𝑡)𝑆₂𝑆(𝑡)₃(𝑡)𝑆₄𝑥(𝑡)𝑦−𝑘₁𝑥𝑦(𝑡)𝒞, (2.72) one gets (𝑎₁∗𝑆₃)(𝑡)𝑥=𝑆₁(𝑡)𝑥−𝑘₁(𝑡)𝐶𝑥,(𝑎₁∗𝑆₄)(𝑡)𝑥=𝑆₂(𝑡)𝑥,𝐴(𝑎₁∗𝑆₃)(𝑡)𝑥=𝑆₃(𝑡)𝑥 and 𝐴(𝑎₁∗𝑆₂)(𝑡)𝑥=𝑆₄(𝑡)𝑥−𝑘₁(𝑡)𝐶𝑥,0≤𝑡<𝜏. Hence, 𝐴(𝑎∗𝑅)(𝑡)𝑥=𝐴(𝑎∗(𝑆₃+𝑘𝐶))(𝑡)𝑥=𝐴(𝑎₁∗𝑎₁∗(𝑆₃+𝑘𝐶))(𝑡)𝑥=𝐴(𝑎₁∗(𝑆₁−𝑘₁𝐶+(𝑎₁∗𝑘)𝐶))(𝑡)𝑥=𝑆₃(𝑡)𝑥=𝑅(𝑡)𝑥−𝑘(𝑡)𝐶𝑥,𝑡∈[0,𝜏). This implies that (𝑅(𝑡))_{𝑡∈[0,𝜏)} is a nondegenerate operator family, and we finally get that (𝑅(𝑡))_{𝑡∈[0,𝜏)} is an (𝑎,𝑘)-regularized 𝐶-resolvent family with a subgenerator 𝐴. The remnant of the proof follows from a slight technical modification of the final part of the proof of [15, Proposition ?3].

Remark 2.35. (i) Let 𝜏=∞, and let 𝑘(𝑡) and 𝑎₁(𝑡) be exponentially bounded. Then (𝑅(𝑡))_{𝑡∈[0,𝜏)} is exponentially bounded if and only if (𝑆(𝑡))_{𝑡∈[0,𝜏)} is exponentially bounded.
(ii) Let 𝑗∈ℕ,𝛼>0,𝑎₁(𝑡)=𝑡^𝛼−1/Γ(𝛼) and 𝑘∈𝐶^𝑗((0,𝜏)), respectivley, 𝑘∈𝐶^∞((0,𝜏)), and let the mapping 𝑡↦𝑅(𝑡),𝑡∈(0,𝜏) be 𝑗-times differentiable, respectivley infinitely differentiable. Then the mapping 𝑡↦𝑆(𝑡),𝑡∈(0,𝜏) is also 𝑗-times differentiable, resp. infinitely differentiable. Furthermore, if 𝑘(𝑡) is of class 𝐶^𝐿, resp. 𝐶_𝐿 (𝜌-hypoanalytic, 1≤𝜌<∞) and (𝑅(𝑡))_{𝑡∈[0,𝜏)} is of class 𝐶^𝐿, resp. 𝐶_𝐿 (𝜌-hypoanalytic), then (𝑆(𝑡))_{𝑡∈[0,𝜏)} is also of class 𝐶^𝐿, resp. 𝐶_𝐿 (𝜌-hypoanalytic).
(iii) Let 𝑎₁(𝑡)=1/√𝜋𝑡 and 𝑘₁(𝑡)=𝑡^𝑛−(1/2)/Γ(𝑛−(1/2)),𝑛∈ℕ. Then Theorem 2.34 enables one to discuss the maximal interval of existence of a local (𝑎₁,𝑘₁)-regularized 𝐶-resolvent family and to construct an example of a local (𝑎₁,𝑘₁)-regularized 𝐶-resolvent family (𝑅(𝑡))_{𝑡∈[0,𝜏)} which cannot be extended beyond the interval [0,𝜏); combining with [25, Examples ?1, ?3, ?5] and [34, Theorem ?3.1], it is possible to construct examples of infinitely differentiable, nonanalytic (𝑎₁,𝑘₁)-regularized 𝐶-resolvent families and examples of (pseudo)differential operators generating (𝑎₁,𝑘₁)-regularized 𝐶-resolvent families of class 𝐶^𝐿.
(iv) Assume 𝑎₁(𝑡)=1/√𝜋𝑡,𝐴 is a subgenerator of a (local) 𝐾-convoluted 𝐶-semigroup (𝑆_𝐾(𝑡))_{𝑡∈[0,𝜏)},𝑘(𝑡)=∫^𝑡₀𝐾(𝑠)𝑑𝑠,𝑡∈[0,𝜏), and (H3) holds (see Theorem 2.2(iii). Let 𝑘₁(𝑡) possess the same meaning as in Theorem 2.34. Then, for every 𝑥∈𝐷(𝐴) and 𝑦∈𝐸, the system of integral equations
𝑢∈𝐶([0,𝜏)∶[𝐷(𝐴)]),𝑣∈𝐶([0,𝜏)∶𝐸),𝑢(𝑡)=𝑘₁(𝑡)𝐶𝑥+^𝑡₀𝑣(𝑠)𝑑𝑠𝜋(𝑡−𝑠),𝑡∈[0,𝜏),𝑣(𝑡)=𝑘₁(𝑡)𝐶𝑦+^𝑡₀𝐴𝑢(𝑠)𝑑𝑠𝜋(𝑡−𝑠),𝑡∈[0,𝜏), (2.73) has a unique solution.