On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator

Abstract

For the evolution equation 𝑦^′(𝑡)=𝐴𝑦(𝑡) with a scalar type spectral operator 𝐴 in a Banach space, conditions on 𝐴 are found that are necessary and sufficient for all weak solutions of the equation on [0,∞) to be strongly infinite differentiable on [0,∞) or [0,∞). Certain effects of smoothness improvement of the weak solutions are analyzed.

1. Introduction

Consider the evolution equation𝑦^′(𝑡)=𝐴𝑦(𝑡) (1.1) with a scalar type spectral operator 𝐴 in a complex Banach space 𝑋.

Following [1], we understand by a weak solution of equation (1.1) on an interval [0,𝑇) (0<𝑇≤∞) such a vector function 𝑦∶[0,𝑇)→𝑋 that (1)𝑦(⋅) is strongly continuous on [0,𝑇); (2)for any 𝑔^∗∈𝐷(𝐴^∗), 𝑑𝑦(𝑡)𝑑𝑡,𝑔^∗=𝑦(𝑡),𝐴^∗𝑔^∗,𝑡∈[0,𝑇). (1.2) (𝐷(⋅) is the domain of an operator, 𝐴^∗ is the operator adjoint to 𝐴, and ⟨⋅,⋅⟩ is the pairing between the space 𝑋 and its dual 𝑋^∗).

As was shown [2], Theorem 4.2, the general weak solution of (1.1) on [0,𝑇) (0<𝑇≤∞) is as follows: 𝑦(𝑡)=𝑒^𝑡𝐴𝑓,𝑡∈[0,𝑇),𝑓∈_{0≤𝑡<𝑇}𝐷𝑒^𝑡𝐴, (1.3) the operator exponentials 𝑒^𝑡𝐴, 0≤𝑡<𝑇, defined in the sense of the operational calculus for scalar type spectral operators (see Section 2).

Here, we are to find conditions on 𝐴 that are necessary and sufficient for all weak solutions of (1.1) on [0,∞) to be strongly infinite differentiable on [0,∞) or (0,∞).

This goal attained; all the principal results of paper [3] and the corresponding ones of [4] obtain their natural generalizations.

2. Preliminaries

Henceforth, unless specified otherwise, let 𝐴 be a scalar type spectral operator in a complex Banach space 𝑋 with a norm ‖⋅‖, 𝐸_𝐴(⋅) its spectral measure (the resolution of the identity), and 𝜎(𝐴) the operator's spectrum, with the latter being the support for the former.

Observe that, in a Hilbert space, the scalar type spectral operators are those similar to the normal ones [5].

For scalar type spectral operators, there has been developed an operational calculus for Borel measurable functions defined on 𝜎(𝐴) [6, 7]. With 𝐹(⋅) being such a function, a new scalar type spectral operator𝐹(𝐴)=^def_𝜎(𝐴)𝐹(𝜆)𝑑𝐸_𝐴(𝜆) (2.1) is defined as follows: 𝐹(𝐴)𝑓=^deflim_𝑛→∞𝐹_𝑛(𝐴)(𝐹(𝐴,𝐷(𝐹(𝐴𝑓,𝑓∈𝐷))))=^def𝑓∈𝑋∣lim_𝑛→∞𝐹_𝑛(𝐴)𝑓exists, (2.2) where 𝐹_𝑛(⋅)=^def𝐹(⋅)𝜒_{{𝜆∈𝜎(𝐴)∣|𝐹(𝜆)|≤𝑛}}(⋅),𝑛=1,2,…, (2.3) (𝜒_𝛼(⋅) is the characteristic function of a set 𝛼) and 𝐹_𝑛(𝐴)=^def_𝜎(𝐴)𝐹_𝑛(𝜆)𝑑𝐸_𝐴(𝜆),𝑛=1,2,…, (2.4) are bounded scalar type spectral operators on 𝑋 defined in the same manner as for normal operators (see, e.g., [8, 9]).

In particular, 𝐴=_𝜎(𝐴)𝜆𝑑𝐸_𝐴(𝜆). (2.5) The properties of the spectral measure, 𝐸_𝐴(⋅), and the operational calculus underlying our discourse are exhaustively delineated in [6, 7]. We will outline here a few noteworthy facts.

Due to its strong countable additivity, the spectral measure 𝐸_𝐴(⋅) is bounded [10], that is, there is an 𝑀>0 such that𝐸_𝐴(𝛼)≤𝑀,𝛼∈ℬ (2.6) (ℬ is the 𝜎-algebra of Borel sets in the complex plane ℂ).

Observe that ‖⋅‖ is used in (2.6) to designate the norm in the space of bounded linear operators on 𝑋. We will adhere to this rather common economy of symbols adopting the same notation for the norm in the dual space 𝑋^∗ as well.

For any 𝑓∈𝑋 and 𝑔^∗∈𝑋^∗, let 𝑣(𝑓,𝑔^∗,⋅) be the total variation of the complex-valued measure ⟨𝐸_𝐴(⋅)𝑓,𝑔^∗⟩ on ℬ.

As we discussed in [4, 11], with 𝐹(⋅) being an arbitrary Borel measurable function on ℂ, for any 𝑓∈𝐷(𝐹(𝐴)), 𝑔^∗∈𝑋^∗, and 𝛿∈ℬ,_𝛿|𝐹(𝜆)|𝑑𝑣𝑓,𝑔^∗,𝜆≤4𝑀𝐸_𝐴(𝛿)𝐹(𝐴)𝑓𝑔^∗, (2.7) where 𝑀>0 is the constant from (2.6).

In particular, for 𝐹(𝜆)=1, 𝜆∈ℂ, and 𝛿=ℂ (or any Borel set containing 𝜎(𝐴)), we have𝑣𝑓,𝑔^∗,𝛿≤4𝑀𝑓𝑔^∗,𝑓∈𝑋,𝑔^∗∈𝑋^∗. (2.8) Further, for a Borel measurable nonnegative function 𝐹(⋅) on ℂ, a 𝛿∈ℬ, and a sequence {Δ_𝑛}^∞_𝑛=1 of pairwise disjoint Borel sets in ℂ,_𝛿𝐹(𝜆)𝑑𝑣𝐸_𝐴∪^∞_𝑛=1Δ_𝑛𝑓,𝑔^∗=,𝜆_{∞𝑛=1𝛿∩Δ𝑛}𝐹(𝜆)𝑑𝑣𝐸_𝐴Δ_𝑛𝑓,𝑔^∗,𝜆,𝑓∈𝑋,𝑔^∗∈𝑋^∗. (2.9) Indeed, since for the spectral measure [6, 7] 𝐸_𝐴(𝛼)𝐸_𝐴𝛽=𝐸_𝐴𝛼∩𝛽,𝛼,𝛽∈ℬ, (2.10) for the total variation, we have𝑣𝐸_𝐴𝛽𝑓,𝑔^∗,𝛼=𝑣𝑓,𝑔^∗,𝛼∩𝛽,𝛼,𝛽∈ℬ. (2.11) Whence, due to nonnegativity of 𝐹(⋅) [12], _𝛿𝐹(𝜆)𝑑𝑣𝐸_𝐴∪^∞_𝑛=1Δ_𝑛𝑓,𝑔^∗=,𝜆_{𝛿∩∪∞𝑛=1Δ𝑛}𝐹(𝜆)𝑑𝑣𝑓,𝑔^∗,𝜆=^∞_{𝑛=1𝛿∩Δ𝑛}𝐹(𝜆)𝑑𝑣𝑓,𝑔^∗=,𝜆_{∞𝑛=1𝛿∩Δ𝑛}𝐹(𝜆)𝑑𝑣𝐸_𝐴Δ_𝑛𝑓,𝑔^∗,𝜆. (2.12) We shall need the following regions in the complex plane: ℒ=_𝑏+^def𝜆∈ℂ∣Re𝜆≥max0,𝑏₊||,ℒlnIm𝜆_{𝑏−,𝑏+def}=𝜆∈ℂ∣Re𝜆≤min0,−𝑏₋ln|Im𝜆|orRe𝜆≥max0,𝑏₊ln|Im𝜆|, (2.13) where 𝑏₊ and 𝑏₋ are positive constants.

The terms spectral measure and operational calculus frequently referred to will be abbreviated to s.m. and o.c., respectively.

3. Differentiability of a Particular Weak Solution

Proposition 3.1. Let 𝑛=1,2,… and 𝐼 be a subinterval of an interval [0,𝑇)(0<𝑇≤∞). A weak solution 𝑦(⋅) of (1.1) on [0,𝑇) is 𝑛 times strongly differentiable on 𝐼 if and only if 𝑦(𝑡)∈𝐷𝐴^𝑛,𝑡∈𝐼, (3.1) in which case, 𝑦^(𝑘)(𝑡)=𝐴^𝑘𝑦(𝑡),𝑘=1,2,…,𝑛,𝑡∈𝐼. (3.2)

Proof. “Only if” part.
Let 𝑛=1,2,… and suppose that a weak solution 𝑦(⋅) of (1.1) on [0,𝑇) is 𝑛 times strongly differentiable on a subinterval 𝐼⊆[0,𝑇).
Then for any 𝑔^∗∈𝐷(𝐴^∗), 𝑦^′(𝑡),𝑔^∗=𝑑𝑦(𝑡)𝑑𝑡,𝑔^∗=𝑦(𝑡),𝐴^∗𝑔^∗,𝑡∈𝐼. (3.3) Hence, by the closedness of the operator 𝐴 (cf. [1]), 𝑦(𝑡)∈𝐷(𝐴),𝑦^′(𝑡)=𝐴𝑦(𝑡),𝑡∈𝐼. (3.4) This concludes proving the “only if” part for 𝑛=1, in which case, as is to be noted, the subinterval 𝐼 can shrink to a single point 𝑡∈[0,𝑇).
Let 𝑛≥2 and let the interval 𝐼 be not a singleton. Then differentiating (3.4) at an arbitrary 𝑡∈𝐼, we obtain 𝑦^′′(𝑡)=lim_Δ𝑡→0𝑦^′(𝑡+Δ𝑡)−𝑦^′(𝑡)Δ𝑡=lim_Δ𝑡→0𝐴𝑦(𝑡+Δ𝑡)−𝑦(𝑡),Δ𝑡 (3.5) with the increments Δ𝑡 being such that 𝑡+Δ∈𝐼.
Since lim_Δ𝑡→0𝑦(𝑡+Δ𝑡)−𝑦(𝑡)Δ𝑡=𝑦^′(𝑡),𝑡∈𝐼, (3.6) by the closedness of 𝐴, we infer that 𝑦^′(𝑡)∈𝐷(𝐴),𝑦^′′(𝑡)=𝐴𝑦^′(𝑡),𝑡∈𝐼. (3.7) Considering (3.4) and (3.7), 𝑦(𝑡)∈𝐷𝐴²,𝑦^′′(𝑡)=𝐴²𝑦(𝑡),𝑡∈𝐼. (3.8) Continuing inductively in this manner, we arrive at 𝑦(𝑡)∈𝐷𝐴^𝑘,𝑦^(𝑘)(𝑡)=𝐴^𝑘𝑦(𝑡),𝑘=1,2,…,𝑛,𝑡∈𝐼. (3.9) “If” part.
Let 𝑦(⋅) be a weak solution of (1.1) on an interval [0,𝑇) (0<𝑇≤∞) such that for an 𝑛=1,2,… and a subinterval 𝐼⊆[0,𝑇), 𝑦(𝑡)∈𝐷𝐴^𝑛,𝑡∈𝐼. (3.10) As was discussed (see Section 1), 𝑦(𝑡)=𝑒^𝑡𝐴𝑓,𝑡∈[0,𝑇), (3.11) with some 𝑓∈⋂_{0≤𝑡<𝑇}𝐷(𝑒^𝑡𝐴).
The fact that 𝑒^𝑡𝐴𝑓∈𝐷(𝐴^𝑛), 𝑡∈𝐼, implies by the properties of the o.c. and [2], Proposition 3.1, that, for any 𝑔^∗∈𝑋^∗, _𝜎(𝐴)|𝜆|^𝑘𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆<∞,𝑘=1,…,𝑛,𝑡∈𝐼. (3.12) Given a 𝑘=1,…,𝑛 and an arbitrary 𝑡∈𝐼, let us choose a segment [𝑎,𝑏]⊂𝐼 (𝑎<𝑏) so that 𝑡 is its midpoint if 0<𝑡<𝑇 or 𝑎=0 if 𝑡=0. For increments Δ𝑡 such that 𝑎≤𝑡+Δ𝑡≤𝑏 and any 𝑔^∗∈𝑋^∗, we have 𝐴^𝑘−1𝑦(𝑡+Δ𝑡)−𝐴^𝑘−1𝑦(𝑡)Δ𝑡−𝐴^𝑘𝑦(𝑡),𝑔^∗=bythepropertiesofthe𝑜.𝑐.;_𝜎(𝐴)𝜆^𝑘−1𝑒^{(𝑡+Δ𝑡)𝜆}−𝜆^𝑘−1𝑒^𝑡𝜆Δ𝑡−𝜆^𝑘𝑒^𝑡𝜆𝑑𝐸_𝐴(𝜆)𝑓,𝑔^∗=bythepropertiesofthe𝑜.𝑐.;_𝜎(𝐴)𝜆^𝑘−1𝑒^{(𝑡+Δ𝑡)𝜆}−𝜆^𝑘−1𝑒^𝑡𝜆Δ𝑡−𝜆^𝑘𝑒^𝑡𝜆𝑑𝐸_𝐴(𝜆)𝑓,𝑔^∗≤_𝜎(𝐴)𝜆^𝑘−1𝑒^{(𝑡+Δ𝑡)𝜆}−𝜆^𝑘−1𝑒^𝑡𝜆Δ𝑡−𝜆^𝑘𝑒^𝑡𝜆𝑑𝑣𝑓,𝑔^∗,𝜆bythe𝐿𝑒𝑏𝑒𝑠𝑔𝑢𝑒𝐷𝑜𝑚𝑖𝑛𝑎𝑡𝑒𝑑𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒𝑇ℎ𝑒𝑜𝑟𝑒𝑚;⟶0asΔ𝑡⟶0. (3.13) Indeed, for any 𝑘=1,…,𝑛 and an arbitrary 𝜆∈𝜎(𝐴), 𝜆^𝑘−1𝑒^{(𝑡+Δ𝑡)𝜆}−𝜆^𝑘−1𝑒^𝑡𝜆Δ𝑡−𝜆^𝑘𝑒^𝑡𝜆≤𝜆^𝑘−1𝑒^{(𝑡+Δ𝑡)𝜆}−𝜆^𝑘−1𝑒^𝑡𝜆+𝜆Δ𝑡^𝑘𝑒^𝑡𝜆|𝜆|bythe𝑇𝑜𝑡𝑎𝑙𝐶ℎ𝑎𝑛𝑔𝑒𝑇ℎ𝑒𝑜𝑟𝑒𝑚;≤2^𝑘max_{𝑎≤𝑠≤𝑏}𝑒^𝑠Re𝜆|𝜆|≤2^𝑘𝑒^𝑎Re𝜆|𝜆|ifRe𝜆<0,_𝑘𝑒^𝑏Re𝜆()ifRe𝜆≥0,by3.12,consideringthat𝑎,𝑏∈𝐼;∈ℒ¹𝜎(𝐴),𝑣𝑓,𝑔^∗,𝜆,⋅_𝑘−1𝑒^{(𝑡+Δ𝑡)𝜆}−𝜆^𝑘−1𝑒^𝑡𝜆Δ𝑡−𝜆^𝑘𝑒^𝑡𝜆⟶0asΔ𝑡→0. (3.14)
Therefore, for any 𝑔^∗∈𝑋^∗, 𝑑^𝑘𝑑𝑡^𝑘𝑦(𝑡),𝑔^∗=𝐴^𝑘𝑦(𝑡),𝑔^∗,𝑘=1,…,𝑛,𝑡∈𝐼. (3.15) For Δ_𝑚∶={𝜆∈ℂ∣|𝜆|≤𝑚}, 𝑚=1,2,…, let us fix an arbitrary 𝑘=1,…,𝑛 and consider the sequence of vector functions 𝑦_𝑚(𝑡)∶=𝐸_𝐴Δ_𝑚𝐴^𝑘𝑦(𝑡)=𝐸_𝐴Δ_𝑚𝐴^𝑘𝑒^𝑡𝐴𝑓,𝑚=1,2,…,𝑡∈𝐼. (3.16) Since 𝑓∈𝐷(𝐴^𝑘𝑒^𝑡𝐴), 𝑡∈𝐼, by the properties of the o.c., 𝐸_𝐴Δ_𝑚𝐴^𝑘𝑒^𝑡𝐴𝑓=𝐴𝐸_𝐴Δ_𝑚^𝑘𝑒^{𝑡𝐴𝐸𝐴(Δ𝑚)}𝑓,𝑚=1,2,…,𝑡∈𝐼. (3.17) Due to the boundedness of Δ_𝑚, 𝑚=1,2,…, by the properties of the o.c., 𝐴𝐸_𝐴(Δ_𝑚), 𝑚=1,2,…, is a bounded linear operator on 𝑋 (‖𝐴𝐸_𝐴(Δ_𝑚)‖≤4𝑀𝑚, 𝑚=1,2,…, [7]). Hence, the vector functions 𝑦_𝑚(𝑡)=𝐸_𝐴Δ_𝑚𝐴^𝑘𝑒^𝑡𝐴𝑓,𝑚=1,2,…,𝑡∈𝐼, (3.18) are strongly continuous on 𝐼.
For an arbitrary segment [𝑎,𝑏]⊆𝐼, we have sup_{𝑎≤𝑡≤𝑏}𝐴^𝑘𝑒^𝑡𝐴𝑓−𝐸_𝐴Δ_𝑚𝐴^𝑘𝑒^𝑡𝐴𝑓bythepropertiesofthe𝑜.𝑐.;=sup_{𝑎≤𝑡≤𝑏{(𝐴)∣|𝜆|}𝜆∈𝜎>𝑚}𝜆^𝑘𝑒^𝑡𝜆𝑑𝐸_𝐴(𝜆)𝑓asfollowsfromthe𝐻𝑎ℎ𝑛-𝐵𝑎𝑛𝑎𝑐ℎ𝑇ℎ𝑒𝑜𝑟𝑒𝑚;=sup_{𝑎≤𝑡≤𝑏}sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1{(𝐴)∣|𝜆|}𝜆∈𝜎>𝑚}𝜆^𝑘𝑒^𝑡𝜆𝑑𝐸_𝐴(𝜆)𝑓,𝑔^∗bythepropertiesofthe𝑜.𝑐.;=sup_{𝑎≤𝑡≤𝑏}sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1{(𝐴)∣|𝜆|}𝜆∈𝜎>𝑚}𝜆^𝑘𝑒^𝑡𝜆𝑑𝐸_𝐴(𝜆)𝑓,𝑔^∗≤sup_{𝑎≤𝑡≤𝑏}sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1{(𝐴)∣|𝜆|}𝜆∈𝜎>𝑚}|𝜆|^𝑘𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆≤sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1}sup_{𝑎≤𝑡≤𝑏{(𝐴)∣|𝜆|}𝜆∈𝜎>𝑚,Re𝜆≤0}|𝜆|^𝑘𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗+,𝜆_{{(𝐴)∣|𝜆|}𝜆∈𝜎>𝑚,Re𝜆>0}|𝜆|^𝑘𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆≤sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1{(𝐴)∣|𝜆|}𝜆∈𝜎>𝑚,Re𝜆≤0}|𝜆|^𝑘𝑒^𝑎Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆+sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1{(𝐴)∣|𝜆|}𝜆∈𝜎>𝑚,Re𝜆>0}|𝜆|^𝑚𝑒^𝑏Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆≤sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1{(𝐴)∣|𝜆|}𝜆∈𝜎>𝑚}|𝜆|^𝑘𝑒^𝑎Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆+sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1{(𝐴)∣|𝜆|}𝜆∈𝜎>𝑚}|𝜆|^𝑘𝑒^𝑏Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆by(2.7);≤4𝑀sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1}𝐸_𝐴({𝜆∈𝜎(𝐴)∣|𝜆|>𝑚})𝐴^𝑘𝑒^𝑎𝐴𝑓𝑔^∗+sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1}𝐸_𝐴({𝜆∈𝜎(𝐴)∣|𝜆|>𝑚})𝐴^𝑘𝑒^𝑏𝐴𝑓𝑔^∗=4𝑀𝐸_𝐴({𝜆∈𝜎(𝐴)∣|𝜆|>𝑚})𝐴^𝑘𝑒^𝑎𝐴𝑓+𝐸_𝐴({𝜆∈𝜎(𝐴)∣|𝜆|>𝑚})𝐴^𝑘𝑒^𝑏𝐴𝑓bythestrongcontinuityofthe𝑠.𝑚.;⟶0as𝑚⟶∞. (3.19) Hence, for any 𝑘=1,…,𝑛, the vector function 𝐴^𝑘𝑒^𝑡𝐴𝑓, 𝑡∈𝐼, is strongly continuous on 𝐼, being the uniform limit of the sequence of strongly continuous on 𝐼 vector functions {𝑦_𝑚(⋅)}^∞_𝑚=1 on any segment [𝑎,𝑏]⊆𝐼.
Let us fix an arbitrary 𝑎∈𝐼 and integrate (3.15) for 𝑘=1 between 𝑎 and an arbitrary 𝑡∈𝐼. Considering the strong continuity of 𝐴𝑒^𝑡𝐴𝑓, 𝑡∈𝐼, we have 𝑒^𝑡𝐴𝑓−𝑒^𝑎𝐴𝑓,𝑔^∗=^𝑡_𝑎𝐴𝑒^𝑠𝐴𝑓𝑑𝑠,𝑔^∗,𝑔^∗∈𝑋^∗. (3.20) Whence, as follows from the Hahn-Banach Theorem, 𝑒^𝑡𝐴𝑓−𝑒^𝑎𝐴𝑓=^𝑡_𝑎𝐴𝑒^𝑠𝐴𝑓𝑑𝑠,𝑡∈𝐼. (3.21) By the strong continuity of 𝐴𝑒^𝑡𝐴𝑓, 𝑡∈𝐼, 𝑑𝑒𝑑𝑡^𝑡𝐴𝑓=𝐴𝑒^𝑡𝐴𝑓,𝑡∈𝐼. (3.22) Consequently, by (3.15), for 𝑛=2, 𝑑𝑑𝑑𝑡𝑒𝑑𝑡^𝑡𝐴𝑓,𝑔^∗=𝐴²𝑒^𝑡𝐴𝑓,𝑔^∗,𝑡∈𝐼. (3.23) Whence, analogously, 𝑑²𝑑𝑡²𝑒^𝑡𝐴𝑓=𝐴²𝑒^𝑡𝐴𝑓,𝑡∈𝐼. (3.24) Continuing inductively in this manner, we infer that, for any 𝑘=1,…,𝑛, 𝑑^𝑘𝑑𝑡^𝑘𝑒^𝑡𝐴𝑓=𝐴^𝑘𝑒^𝑡𝐴𝑓,𝑡∈𝐼. (3.25)

Corollary 3.2. Let 𝐼 be a subinterval of an interval [0,𝑇)(0<𝑇≤∞). A weak solution 𝑦(⋅) of (1.1) on [0,𝑇) is strongly infinite differentiable on 𝐼 if and only if 𝑦(𝑡)∈𝐶^∞(𝐴)=^def∞_𝑛=1𝐷𝐴^𝑛,𝑡∈𝐼, (3.26) in which case, 𝑦^(𝑛)(𝑡)=𝐴^𝑛𝑦(𝑡),𝑛=1,2,…,𝑡∈𝐼. (3.27)

Thus, we have obtained generalizations of Proposition 4.1 and Corollary 4.1 of [3], respectively.

4. Differentiability of Weak Solutions

Theorem 4.1. Every weak solution of (1.1) on [0,∞) is strongly infinite differentiable on [0,∞) if and only if there is a 𝑏₊>0 such that the set 𝜎(𝐴)⧵ℒ_𝑏+ is bounded.

Proof. “If” part.
Let 𝑏₊>0 be such that the set 𝜎(𝐴)⧵ℒ_𝑏+ is bounded and 𝑦(⋅) a weak solution of (1.1) on [0,∞). Then (see Section 1) 𝑦(𝑡)=𝑒^𝑡𝐴𝑓,0≤𝑡<∞, (4.1) with some 𝑓∈⋂_0≤𝑡<∞𝐷(𝑒^𝑡𝐴).
For any 𝑛=1,2,…, 𝑡≥0 and an arbitrary 𝑔^∗∈𝑋^∗, _𝜎(𝐴)|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗=,𝜆_{𝜎(𝐴)⧵ℒ𝑏+}|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆+_{𝜎(𝐴)∩ℒ𝑏+}|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆<∞. (4.2) Indeed, the former integral is finite due to the boundedness of the set 𝜎(𝐴)⧵ℒ_𝑏+, the finiteness of the measure 𝑣(𝑓,𝑔^∗,⋅) and the continuity of the integrated function on ℂ.
For the latter one, we have _{𝜎(𝐴)∩ℒ𝑏+}|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗≤,𝜆_{𝜎(𝐴)∩ℒ𝑏+}[|Re𝜆|+|Im𝜆|]^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆for𝜆∈𝜎(A)∩ℒ_𝑏+,Re𝜆≥0,|Im𝜆|≤𝑒^{𝑏+−1Re𝜆};≤_{𝜎(𝐴)∩ℒ𝑏+}Re𝜆+𝑒^{𝑏+−1Re𝜆𝑛}𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆since𝑥≤𝑒^𝑥≤,𝑥≥0;_{𝜎(𝐴)∩ℒ𝑏+}𝑏₊𝑒^{𝑏+−1Re𝜆}+𝑒^{𝑏+−1Re𝜆𝑛}𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗=𝑏,𝜆₊+1^𝑛_𝜎(𝐴)𝑒^{[𝑛𝑏+−1+𝑡]Re𝜆}𝑑𝑣𝑓,𝑔^∗,𝜆since𝑓∈_0≤𝑡<∞𝐷𝑒^𝑡𝐴,by[2],Proposition3.1;<∞. (4.3) Further, for any 𝑛=1,2,… and 𝑡≥0, sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1(𝐴)∣|𝜆|𝜆∈𝜎𝑛𝑒𝑡Re𝜆>𝑚}|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆≤sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1(𝐴)𝜆∈𝜎⧵ℒ𝑏+∣|𝜆|𝑛𝑒𝑡Re𝜆>𝑚}|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆+sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1(𝐴)𝜆∈𝜎∩ℒ𝑏+∣|𝜆|𝑛𝑒𝑡Re𝜆>𝑚}|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆⟶0as𝑚⟶∞. (4.4) Indeed, since, due to the boundedness of the set 𝜎(𝐴)⧵ℒ_𝑏+, the set {𝜆∈𝜎(𝐴)⧵ℒ_𝑏+∣|𝜆|^𝑛𝑒^𝑡Re𝜆>𝑚} is void for all sufficiently large 𝑚's, sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1(𝐴)𝜆∈𝜎⧵ℒ𝑏+∣|𝜆|𝑛𝑒𝑡Re𝜆>𝑚}|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆⟶0as𝑚→∞. (4.5) In addition to this sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1(𝐴)𝜆∈𝜎∩ℒ𝑏+∣|𝜆|𝑛𝑒𝑡Re𝜆>𝑚}|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆analogouslyto(4.3);≤sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1}𝑏₊+1^𝑛_{(𝐴)𝜆∈𝜎∩ℒ𝑏+∣|𝜆|𝑛𝑒𝑡Re𝜆>𝑚}𝑒^{[𝑛𝑏+−1+𝑡]Re𝜆}𝑑𝑣𝑓,𝑔^∗,𝜆since𝑓∈_0≤𝑡<∞𝐷𝑒^𝑡𝐴();≤𝑏,by2.7₊+1^𝑛sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1}4𝑀𝐸_𝐴𝜆∈𝜎(𝐴)∩ℒ_𝑏+∣|𝜆|^𝑛𝑒^𝑡Re𝜆𝑒>𝑚_{(𝑛𝑏+−1+𝑡)𝐴}𝑓𝑔^∗=4𝑀𝑏₊+1^𝑛𝐸_𝐴𝜆∈𝜎(𝐴)∩ℒ_𝑏+∣|𝜆|^𝑛𝑒^𝑡Re𝜆>𝑚𝑒^{(𝑛𝑏+−1+𝑡)𝐴}𝑓bythestrongcontinuityofthe𝑠.𝑚.;⟶0as𝑚⟶∞. (4.6)
By the properties of the o.c. and [2], Proposition 3.1, (4.2) and (4.4) imply that 𝑦(𝑡)=𝑒^𝑡𝐴𝑓∈𝐶^∞(𝐴),0≤𝑡<∞. (4.7)
Then, by Corollary 3.2, 𝑦(⋅) is strongly infinite differentiable on [0,∞).
“Only if” part.
We will prove this part by contrapositive.
Assume that, for any 𝑏₊>0, the set 𝜎(𝐴)⧵ℒ_𝑏+ is unbounded.
In particular, for any 𝑛=1,2,…, unbounded is the set 𝜎(𝐴)⧵ℒ_(2𝑛)⁻¹.
Hence, we can choose a sequence of points in the complex plane {𝜆_𝑛}^∞_𝑛=1 as follows: 𝜆_𝑛∈𝜎(𝐴),𝑛=1,2,…,Re𝜆_𝑛<max0,(2𝑛)⁻¹||𝜆lnIm𝜆,𝑛=1,2,…,₀∶=0,𝜆_𝑛>max𝑛⁴,𝜆_𝑛−1,𝑛=1,2,…. (4.8)
The latter implies in particular that the points 𝜆_𝑛 are distinct (𝜆_𝑖≠𝜆_𝑗, 𝑖≠𝑗).
Since, for any 𝑛=1,2,…, the set 𝜆∈ℂ∣Re𝜆<max0,(2𝑛)⁻¹ln|Im𝜆|,|𝜆|>max𝑛⁴,𝜆_𝑛−1 (4.9) is open in ℂ, there exists an 𝜀_𝑛>0 such that this set along with the point 𝜆_𝑛 contains the open disk of radius 𝜀_𝑛 centered at 𝜆_𝑛Δ_𝑛∶=𝜆∈ℂ∣𝜆−𝜆_𝑛<𝜀_𝑛. (4.10) Then, for any 𝜆∈Δ_𝑛, 𝑛=1,2,…, Re𝜆<max0,(2𝑛)⁻¹ln|Im𝜆|,|𝜆|>max𝑛⁴,𝜆_𝑛−1. (4.11) Further, since the points 𝜆_𝑛, 𝑛=1,2,…, are distinct, we can regard the radii of the disks, 𝜀_𝑛, 𝑛=1,2,…, to be small enough so that 0<𝜀_𝑛<1𝑛Δ,𝑛=1,2,…,_𝑖∩Δ_𝑗=∅,𝑖≠𝑗i.e.,thedisksare𝑝𝑎𝑖𝑟𝑤𝑖𝑠𝑒𝑑𝑖𝑠𝑗𝑜𝑖𝑛𝑡. (4.12) Whence, by the properties of the s.m., 𝐸_𝐴Δ_𝑖𝐸_𝐴Δ_𝑗=0,𝑖≠𝑗. (4.13) Observe also, that the subspaces 𝐸_𝐴(Δ_𝑛)𝑋, 𝑛=1,2,…, are nontrivial since Δ_𝑛∩𝜎(𝐴)≠∅, with Δ_𝑛 being an open set.
By choosing a unit vector 𝑒_𝑛∈𝐸_𝐴(Δ_𝑛)𝑋 (‖𝑒_𝑛‖=1), 𝑛=1,2,…, we obtain a vector sequence such that 𝐸_𝐴Δ_𝑖𝑒_𝑗=𝛿_𝑖𝑗𝑒_𝑖,𝑖,𝑗=1,2,… (4.14) (𝛿_𝑖𝑗 is the Kronecker delta symbol).
As is readily verified, (4.14) implies that the vectors {𝑒₁,𝑒₂,…} are linearly independent.
Moreover, there is an 𝜀>0 such that 𝑑_𝑛∶=dist𝑒_𝑛,span𝑒_𝑘∣𝑘=1,2,…,𝑘≠𝑛≥𝜀,𝑛=1,2,…. (4.15) Indeed, the opposite implies the existence of a subsequence {𝑑_𝑛(𝑘)}^∞_𝑘=1 such that 𝑑_𝑛(𝑘)⟶0as𝑘⟶∞. (4.16) Then, for any 𝑘=1,2,…, by selecting a vector 𝑓_𝑛(𝑘)∈span({𝑒_𝑗∣𝑗=1,2,…,𝑗≠𝑛(𝑘)}) such that ‖𝑒_𝑛(𝑘)−𝑓_𝑛(𝑘)‖<𝑑_𝑛(𝑘)+1/𝑘, considering (4.14) and (2.6), we arrive at the contradiction: 1=𝑒_𝑛(𝑘)=𝐸_𝐴Δ_𝑛(𝑘)𝑒_𝑛(𝑘)−𝑓_𝑛(𝑘)≤𝑀𝑒_𝑛(𝑘)−𝑓_𝑛(𝑘)⟶0as𝑘⟶∞. (4.17) As follows from the Hahn-Banach Theorem, for any 𝑛=1,2,…, there is an 𝑒^∗_𝑛∈𝑋^∗ such that 𝑒^∗_𝑛=1,𝑒_𝑖,𝑒^∗_𝑗=𝛿_𝑖𝑗𝑑_𝑖,𝑖,𝑗=1,2,…. (4.18) Concerning the sequence of the real parts, {Re𝜆_𝑛}^∞_𝑛=1, there are two possibilities: it is either bounded or unbounded above.
Suppose that the sequence {Re𝜆_𝑛}^∞_𝑛=1 is bounded above, that is, there is such an 𝜔>0 that Re𝜆_𝑛≤𝜔,𝑛=1,2,…. (4.19) Let 𝑓∶=^∞_𝑛=11𝑛²𝑒_𝑛. (4.20) By (4.14), 𝐸_𝐴∪^∞_𝑛=1Δ_𝑛𝑓=𝑓,𝐸_𝐴Δ_𝑛1𝑓=𝑛²𝑒_𝑛,𝑛=1,2,…. (4.21) For any 𝑡≥0 and an arbitrary 𝑔^∗∈𝑋^∗, _𝜎(𝐴)𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗();=,𝜆by4.21_𝜎(𝐴)𝑒^𝑡Re𝜆𝑑𝑣𝐸_𝐴∪^∞_𝑛=1Δ_𝑛𝑓,𝑔^∗();=,𝜆by2.9_{∞𝑛=1Δ𝑛}𝑒^𝑡Re𝜆𝑑𝑣𝐸_𝐴Δ_𝑛𝑓,𝑔^∗();=,𝜆by4.21_∞𝑛=11𝑛²_Δ𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑒_𝑛,𝑔^∗,𝜆for𝜆∈Δ_𝑛,by(4.12),and(4.19),Re𝜆=Re𝜆_𝑛+Re𝜆−Re𝜆_𝑛≤Re𝜆_𝑛+𝜆−𝜆_𝑛≤𝜔+𝜀_𝑛≤𝜔+1;≤𝑒^{∞𝑡(𝜔+1)}_𝑛=11𝑛²𝑣𝑒_𝑛,𝑔^∗,Δ_𝑛by(2.8);≤𝑒^{∞𝑡(𝜔+1)}_𝑛=11𝑛²4𝑀𝑒_𝑛𝑔^∗=4𝑀𝑒^𝑡(𝜔+1)𝑔^∗∞_𝑛=11𝑛²<∞. (4.22) Similarly, for any 𝑡≥0, sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1(𝐴)𝜆∈𝜎∣𝑒𝑡Re𝜆>𝑛}𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆≤sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1}𝑒^{∞𝑡(𝜔+1)}_𝑛=11𝑛²_{Δ𝑛∩(𝐴)𝜆∈𝜎∣𝑒}^𝑡Re𝜆_>𝑛1𝑑𝑣𝑒_𝑛,𝑔^∗,𝜆by(2.9)and(4.21);=𝑒^𝑡(𝜔+1)sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1(𝐴)𝜆∈𝜎∣𝑒𝑡Re𝜆>𝑛}1𝑣𝑓,𝑔^∗,𝜆by(2.7);≤𝑒^𝑡(𝜔+1)sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1}4𝑀𝐸_𝐴𝜆∈𝜎(𝐴)∣𝑒^𝑡Re𝜆>𝑛𝑓𝑔^∗=4𝑀𝑒^𝑡(𝜔+1)𝐸_𝐴𝜆∈𝜎(𝐴)∣𝑒^𝑡Re𝜆>𝑛𝑓bythestrongcontinuityofthe𝑠.𝑚.;⟶0as𝑛⟶∞. (4.23) By [2], Proposition 3.1, (4.22) and (4.23) imply that 𝑓∈⋂_0≤𝑡<∞𝐷(𝑒^𝑡𝐴).
Therefore (see Section 1), 𝑦(𝑡)∶=𝑒^𝑡𝐴𝑓, 0≤𝑡<∞, is a weak solution of (1.1) on [0,∞).
Let ℎ^∗∶=^∞_𝑛=11𝑛²𝑒^∗_𝑛∈𝑋^∗, (4.24) (cf. (4.18)).
Considering (4.18) and (4.15), we have 𝑒_𝑛,ℎ^∗=𝑑_𝑛𝑛²≥𝜀𝑛²,𝑛=1,2,…. (4.25) Similarly to (4.22), _𝜎(𝐴)|𝜆|𝑑𝑣𝑓,ℎ^∗,𝜆=^∞_𝑛=11𝑛²_Δ𝑛|𝜆|𝑑𝑣𝑒_𝑛,ℎ^∗,𝜆for𝜆∈Δ_𝑛,by(4.11),|𝜆|≥𝑛⁴;≥^∞_𝑛=1𝑛²𝑣𝑒_𝑛,ℎ^∗,Δ_𝑛≥^∞_𝑛=1𝑛²𝐸_𝐴Δ_𝑛𝑒_𝑛,ℎ^∗()();≥by4.14and4.25_∞𝑛=1𝑛²𝜀𝑛²=∞. (4.26) Whence, by [2], Proposition 3.1, 𝑦(0)=𝑓∉𝐷(𝐴).
Consequently, by Proposition 3.1, the weak solution 𝑦(𝑡)=𝑒^𝑡𝐴𝑓, 0≤𝑡<∞, of (1.1) on [0,∞) is not strongly differentiable at 0.
Now, assume that the sequence {Re𝜆_𝑛}^∞_𝑛=1 is unbounded above.
Therefore, there is a subsequence {Re𝜆_𝑛(𝑘)}^∞_𝑘=1 such that Re𝜆_𝑛(𝑘)≥𝑘,𝑘=1,2,…. (4.27) Then the vector 𝑓∶=^∞_𝑘=1𝑒^{−𝑛(𝑘)Re𝜆𝑛(𝑘)}𝑒_𝑛(𝑘) (4.28) is well defined.
By (4.14), 𝐸_𝐴∪^∞_𝑘=1Δ_𝑛(𝑘)𝐸𝑓=𝑓,_𝐴Δ_𝑛(𝑘)𝑓=𝑒^{−𝑛(𝑘)Re𝜆𝑛(𝑘)}𝑒_𝑛(𝑘),𝑘=1,2,…. (4.29) For any 𝑡≥0 and an arbitrary 𝑔^∗∈𝑋^∗, similar to (4.22), _𝜎(𝐴)𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆=^∞_𝑘=1𝑒^{−𝑛(𝑘)Re𝜆𝑛(𝑘)}_Δ𝑛(𝑘)𝑒^𝑡Re𝜆𝑑𝑣𝑒_𝑛(𝑘),𝑔^∗,𝜆for𝜆∈Δ_𝑛(𝑘),by(4.12),Re𝜆=Re𝜆_𝑛(𝑘)+Re𝜆−Re𝜆_𝑛(𝑘)≤Re𝜆_𝑛(𝑘)+𝜆−𝜆_𝑛(𝑘)≤Re𝜆_𝑛(𝑘)+1;≤𝑒^𝑡∞_𝑘=1𝑒^{−[𝑛(𝑘)−𝑡]Re𝜆𝑛(𝑘)}𝑣𝑒_𝑛(𝑘),𝑔^∗,Δ_𝑛(𝑘)by(2.8);≤𝑒^𝑡∞_𝑘=1𝑒^{−[𝑛(𝑘)−𝑡]Re𝜆𝑛(𝑘)}4𝑀𝑒_𝑛(𝑘)𝑔^∗=4𝑀𝑒^𝑡𝑔^∗∞_𝑘=1𝑒^{−[𝑛(𝑘)−𝑡]Re𝜆𝑛(𝑘)}bythe𝐶𝑜𝑚𝑝𝑎𝑟𝑖𝑠𝑜𝑛𝑇𝑒𝑠𝑡;<∞. (4.30) Indeed, for all sufficiently large 𝑘's, 𝑛(𝑘)−𝑡≥1 and by (4.27), 𝑒^{−[𝑛(𝑘)−𝑡]Re𝜆𝑛(𝑘)}≤𝑒^−𝑘. (4.31) Analogously, for an arbitrary 𝑡≥0, sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1(𝐴)𝜆∈𝜎∣𝑒𝑡Re𝜆>𝑛}𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆≤sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1}𝑒^𝑡∞_𝑘=1𝑒^{−[𝑛(𝑘)−𝑡]Re𝜆𝑛(𝑘)}_{Δ𝑛(𝑘)∩(𝐴)𝜆∈𝜎∣𝑒𝑡Re𝜆>𝑛}1𝑑𝑣𝑒_𝑛(𝑘),𝑔^∗,𝜆=𝑒^𝑡sup_{𝑔∗∈𝑋∗∣‖𝑔∗}^∞_‖=1𝑘=1𝑒^{−[(𝑛(𝑘)/2)−𝑡]Re𝜆𝑛(𝑘)}𝑒^{−(𝑛(𝑘)/2)Re𝜆(𝑘)}_{Δ𝑛(𝑘)∩(𝐴)𝜆∈𝜎∣𝑒𝑡Re𝜆>𝑛}1𝑑𝑣𝑒_𝑛(𝑘),𝑔^∗(),𝜆dueto4.27,thereisanL>0suchthat𝑒^{−[(𝑛(𝑘)/2)−𝑡]Re𝜆𝑛(𝑘)}≤𝐿,𝑘=1,2,…;≤𝐿𝑒^𝑡sup_{𝑔∗∈𝑋∗∣‖𝑔∗}^∞_‖=1𝑘=1𝑒^{−(𝑛(𝑘)/2)Re𝜆𝑛(𝑘)}_{Δ𝑛(𝑘)∩(𝐴)𝜆∈𝜎∣𝑒𝑡Re𝜆>𝑛}1𝑑𝑣𝑒_𝑛(𝑘),𝑔^∗,𝜆forℎ∶=^∞_𝑘=1𝑒^{−(𝑛(𝑘)/2)Re𝜆𝑛(𝑘)}𝑒_𝑛(𝑘),by(4.14)and(2.9);=𝐿𝑒^𝑡_{(𝐴)𝜆∈𝜎∣𝑒𝑡Re𝜆>𝑛}1𝑑𝑣ℎ,𝑔^∗,𝜆by(2.7);≤𝐿𝑒^𝑡sup_{𝑔∗∈∣‖𝑔∗‖=1}4𝑀𝐸_𝐴𝜆∈𝜎(𝐴)∣𝑒^𝑡Re𝜆>𝑛ℎ𝑔^∗=4𝐿𝑀𝑒^𝑡𝐸_𝐴𝜆∈𝜎(𝐴)∣𝑒^𝑡Re𝜆>𝑛ℎbythestrongcontinuityofthe𝑠.𝑚.;⟶0as𝑛⟶∞. (4.32)
From (4.30) and (4.32), by [2], Proposition 3.1, we infer that 𝑓∈⋂_0≤𝑡<∞𝐷(𝑒^𝑡𝐴).
Therefore (see Section 1), 𝑦(𝑡)∶=𝑒^𝑡𝐴𝑓, 0≤𝑡<∞, is a weak solution of (1.1) on [0,∞).
For any 𝜆∈Δ_𝑛(𝑘), 𝑘=1,2,…, by (4.11), (4.12), and (4.27), Re𝜆=Re𝜆_𝑛(𝑘)−Re𝜆_𝑛(𝑘)−Re𝜆≥Re𝜆_𝑛(𝑘)−Re𝜆_𝑛(𝑘)−Re𝜆≥Re𝜆_𝑛(𝑘)−𝜀_𝑛(𝑘)≥Re𝜆_𝑛(𝑘)−1𝑛(𝑘)((𝑘≥𝑘−1≥0,Re𝜆<max0,2𝑛))⁻¹ln|Im𝜆|. (4.33) Hence, for 𝜆∈Δ_𝑛(𝑘), 𝑘=1,2,…, |𝜆|≥|Im𝜆|≥𝑒^{2𝑛(𝑘)Re𝜆}≥𝑒^{2𝑛(𝑘)(Re𝜆𝑛(𝑘)−1/𝑛(𝑘))}. (4.34) Using this estimate, we obtain _𝜎(𝐴)|𝜆|𝑑𝑣𝑓,ℎ^∗,𝜆=^∞_𝑘=1𝑒^{−𝑛(𝑘)Re𝜆𝑛(𝑘)}_Δ𝑛(𝑘)|𝜆|𝑑𝑣𝑒_𝑛(𝑘),ℎ^∗≥,𝜆_∞𝑘=1𝑒^{−𝑛(𝑘)Re𝜆𝑛(𝑘)}𝑒^{2𝑛(𝑘)(Re𝜆𝑛(𝑘)−1/𝑛(𝑘))}𝑣𝑒_𝑛(𝑘),ℎ^∗,Δ_𝑛(𝑘)=𝑒^∞−2_𝑘=1𝑒^{𝑛(𝑘)Re𝜆𝑛(𝑘)}𝐸_𝐴Δ_𝑛(𝑘)𝑒_𝑛(𝑘),ℎ^∗by(4.14),(4.25),and(4.27);≥𝑒⁻²𝜀^∞_𝑘=1𝑒^𝑛(𝑘)𝑛(𝑘)²=∞. (4.35) Whence, by [2], Proposition 3.1, 𝑦(0)=𝑓∉𝐷(𝐴).
Therefore, by Proposition 3.1, the weak solution 𝑦(𝑡)=𝑒^𝑡𝐴𝑓, 0≤𝑡<∞, of (1.1) on [0,∞) is not strongly differentiable at 0.
With every possibility concerning {Re𝜆_𝑛}^∞_𝑛=1 considered, we infer that the opposite to the assumption that, for a certain 𝑏₊>0, the set 𝜎(𝐴)⧵ℒ_𝑏+ is bounded, allows to single out a weak solution of (1.1) on [0,∞) that is not strongly differentiable at 0, much less strongly infinite differentiable on [0,∞).
Thus, the “only if” part has been proved by contrapositive.

Theorem 5.1 of [3] has been generalized.

Theorem 4.2. Every weak solution of (1.1) on [0,∞) is strongly infinite differentiable on (0,∞) if and only if there is a 𝑏₊>0 such that, for any 𝑏₋>0, the set 𝜎(𝐴)⧵ℒ_{𝑏−,𝑏+} is bounded.

Proof. “If” part.
Let 𝑏₊>0 be such that, for an arbitrary 𝑏₋>0, the set 𝜎(𝐴)⧵ℒ_{𝑏−,𝑏+} is bounded.
Let 𝑦(⋅) be a weak solution of (1.1) on [0,∞). Then (see Section 1) 𝑦(𝑡)=𝑒^𝑡𝐴𝑓,0≤𝑡<∞, (4.36) with some 𝑓∈⋂_0≤𝑡<∞𝐷(𝑒^𝑡𝐴).
For any 𝑛=1,2,…, 𝑡>0, and an arbitrary 𝑔^∗∈𝑋^∗, _𝜎(𝐴)|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗=,𝜆_{𝜎(𝐴)⧵ℒ𝑏−+,𝑏}|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆+_{(𝐴)𝜆∈𝜎∩ℒ𝑏−+,𝑏∣Re𝜆≥0}|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗+,𝜆_{(𝐴)𝜆∈𝜎∩ℒ𝑏−+,𝑏∣Re𝜆<0}|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆<∞. (4.37)
The first integral in this sum is finite due the boundedness of the set 𝜎(𝐴)⧵ℒ_{𝑏−,𝑏+}, the finiteness of the measure 𝑣(𝑓,𝑔^∗,⋅), and the continuity of the integrated function on ℂ.
The finiteness of the second integral is proved in absolutely the same manner as for the corresponding integral in the proof of the “if” part of Theorem 4.1.
Finally for the third one, considering that 𝑏₋>0 is arbitrary and setting 𝑏₋∶=𝑛𝑡⁻¹, we have _{(𝐴)𝜆∈𝜎∩ℒ𝑏−+,𝑏∣Re𝜆<0}|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗≤,𝜆_{(𝐴)𝜆∈𝜎∩ℒ𝑏−+,𝑏∣Re𝜆<0}[|Re𝜆|+|Im𝜆|]^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆for𝜆∈𝜎(𝐴)∩ℒ_{𝑏−,𝑏+}withRe𝜆<0,|Im𝜆|≤𝑒^{𝑏−−1[−Re𝜆]};≤_{(𝐴)𝜆∈𝜎∩ℒ𝑏−+,𝑏∣Re𝜆<0}−Re𝜆+𝑒^{𝑏−−1[−Re𝜆]𝑛}𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆since𝑥≤𝑒^𝑥,𝑥≥0;≤_{(𝐴)𝜆∈𝜎∩ℒ𝑏−+,𝑏∣Re𝜆<0}𝑏₋𝑒^{𝑏−−1[−Re𝜆]}+𝑒^{𝑏−−1[−Re𝜆]𝑛}𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗=𝑏,𝜆₋+1^𝑛_{(𝐴)𝜆∈𝜎∩ℒ𝑏−+,𝑏∣Re𝜆<0}𝑒^{[𝑛𝑏−−1−𝑡][−Re𝜆]}𝑑𝑣𝑓,𝑔^∗,𝜆recallthat𝑏₋∶=𝑛𝑡⁻¹;=𝑏₋+1^𝑛_{(𝐴)𝜆∈𝜎∩ℒ𝑏−+,𝑏∣Re𝜆<0}1𝑑𝑣𝑓,𝑔^∗,𝜆≤𝑏₋+1^𝑛𝑣𝑓,𝑔^∗(𝐴)();𝑏,𝜎by2.8≤4𝑀₋+1^𝑛𝑓𝑔^∗<∞. (4.38) Further, for any 𝑛=1,2,… and 𝑡>0, sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1(𝐴)∣|𝜆|𝜆∈𝜎𝑛𝑒𝑡Re𝜆>𝑚}|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆≤sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1(𝐴)𝜆∈𝜎⧵ℒ𝑏−+,𝑏∣|𝜆|𝑛𝑒𝑡Re𝜆>𝑚}|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆+sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1(𝐴)𝜆∈𝜎∩ℒ𝑏−+,𝑏∣|𝜆|𝑛𝑒𝑡Re𝜆>𝑚}|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆⟶0as𝑚⟶∞. (4.39) Indeed, since, due to the boundedness of the set 𝜎(𝐴)⧵ℒ_{𝑏−,𝑏+}, the set {𝜆∈𝜎(𝐴)⧵ℒ_{𝑏−,𝑏+}∣|𝜆|^𝑛𝑒^𝑡Re𝜆>𝑚} is void for all sufficiently large 𝑚's, sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1(𝐴)𝜆∈𝜎⧵ℒ𝑏−+,𝑏∣|𝜆|𝑛𝑒𝑡Re𝜆>𝑚}|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆⟶0as𝑚⟶∞. (4.40) Similarly to the “if” part of Theorem 4.1 and (4.38), we have sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1(𝐴)𝜆∈𝜎∩ℒ𝑏−+,𝑏∣|𝜆|𝑛𝑒𝑡Re𝜆>𝑚}|𝜆|^𝑛𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆≤sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1}𝑏₊+1^𝑛_{(𝐴){𝜆∈𝜎∩ℒ𝑏−+,𝑏|𝜆|∣Re𝜆≥0,𝑛𝑒𝑡Re𝜆>𝑚}}𝑒^{[𝑛𝑏+−1+𝑡]Re𝜆}𝑑𝑣𝑓,𝑔^∗,𝜆+sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1}𝑏₋+1^𝑛_{(𝐴){𝜆∈𝜎∩ℒ𝑏−+,𝑏|𝜆|∣Re𝜆<0,𝑛𝑒𝑡Re𝜆>𝑚}}𝑒^{[𝑛𝑏−−1−𝑡][−Re𝜆]}𝑑𝑣𝑓,𝑔^∗,𝜆with𝑏₋∶=𝑛𝑡⁻¹;=sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1}𝑏₊+1^𝑛_{(𝐴){𝜆∈𝜎∩ℒ𝑏−+,𝑏|𝜆|∣Re𝜆≥0,𝑛𝑒𝑡Re𝜆>𝑚}}𝑒^{[𝑛𝑏+−1+𝑡]Re𝜆}𝑑𝑣𝑓,𝑔^∗,𝜆+sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1}𝑏₋+1^𝑛_{(𝐴){𝜆∈𝜎∩ℒ𝑏−+,𝑏|𝜆|∣Re𝜆<0,𝑛𝑒𝑡Re𝜆>𝑚}}1𝑑𝑣𝑓,𝑔^∗,𝜆since𝑓∈_0≤𝑡<∞𝐷𝑒^𝑡𝐴();≤𝑏,by2.7₊+1^𝑛sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1}4𝑀𝐸_𝐴𝜆∈𝜎(𝐴)∩ℒ_{𝑏−,𝑏+}∣Re𝜆≥0,|𝜆|^𝑛𝑒^𝑡Re𝜆𝑒>𝑚_{[𝑛𝑏+−1+𝑡]𝐴}𝑓𝑔^∗+𝑏₋+1^𝑛sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1}4𝑀𝐸_𝐴𝜆∈𝜎(𝐴)∩ℒ_{𝑏−,𝑏+}∣Re𝜆<0,|𝜆|^𝑛𝑒^𝑡Re𝜆>𝑚𝑓𝑔^∗=4𝑀𝑏₊+1^𝑛𝐸_𝐴𝜆∈𝜎(𝐴)∩ℒ_{𝑏−,𝑏+}∣Re𝜆≥0,|𝜆|^𝑛𝑒^𝑡Re𝜆>𝑚𝑒^{[𝑛𝑏+−1+𝑡]𝐴}𝑓+4𝑀𝑏₋+1^𝑛𝐸_𝐴𝜆∈𝜎(𝐴)∩ℒ_{𝑏−,𝑏+}∣Re𝜆<0,|𝜆|^𝑛𝑒^𝑡Re𝜆>𝑚𝑓bythestrongcontinuityofthe𝑠.𝑚.;⟶0as𝑚⟶∞. (4.41) By the properties of the o.c. and [2], Proposition 3.1, (4.37) and (4.39) imply that 𝑦(𝑡)=𝑒^𝑡𝐴𝑓∈𝐶^∞(𝐴),0<𝑡<∞. (4.42) Whence, by Corollary 3.2, 𝑦(⋅) is strongly infinite differentiable on (0,∞).
“Only if” part.
As well as in Theorem 4.1, we will prove this part by contrapositive.
Thus, we assume that, for any 𝑏₊>0, there is such a 𝑏₋>0 that the set 𝜎(𝐴)⧵ℒ_{𝑏−,𝑏+} is unbounded.
Let us show that this assumption can even be strengthened. To wit: there is such a 𝑏₋>0 that, for any 𝑏₊>0, the set 𝜎(𝐴)⧵ℒ_{𝑏−,𝑏+} is unbounded.
Indeed, there are two possibilities: (1)for a certain 𝑏₋>0, the set {𝜆∈𝜎(𝐴)∣−𝑏₋ln|Im𝜆|<Re𝜆≤0} is unbounded; (2)for any 𝑏₋>0, the set {𝜆∈𝜎(𝐴)∣−𝑏₋ln|Im𝜆|<Re𝜆≤0} is bounded.
In the first case, the set 𝜎(𝐴)⧵ℒ_{𝑏−,𝑏+} is unbounded with the very 𝑏₋>0, for which the set {𝜆∈𝜎(𝐴)∣−𝑏₋ln|Im𝜆|<Re𝜆≤0} is unbounded, and an arbitrary 𝑏₊>0.
In the second case, based on the premise we infer that, for any 𝑏₊>0, the set {𝜆∈𝜎(𝐴)∣0<Re𝜆<𝑏₊ln|Im𝜆|} is unbounded. Then so is the set 𝜎(𝐴)⧵ℒ_{𝑏−,𝑏+} for any 𝑏₋>0 and 𝑏₊>0.
Thus, let us fix a 𝑏₋>0 such that the set 𝜎(𝐴)⧵ℒ_{𝑏−,𝑏+} is unbounded for an arbitrary 𝑏₊>0.
In particular, for any 𝑛=1,2,…, the set 𝜎(𝐴)⧵ℒ_{𝑏−,(2𝑛)}⁻¹ is unbounded.
Hence, we can select a sequence of points in the complex plane {𝜆_𝑛}^∞_𝑛=1 in the following manner: 𝜆_𝑛∈𝜎(𝐴),𝑛=1,2,…,min0,−𝑏₋ln|Im𝜆|<Re𝜆_𝑛<max0,(2𝑛)⁻¹||𝜆lnIm𝜆𝑛=1,2,…,₀∶=0,𝜆_𝑛>max𝑛⁴,𝜆_𝑛−1,𝑛=1,2,…. (4.43) The latter, in particular, implies that the points 𝜆_𝑛 are distinct.
Since, for any 𝑛=1,2,…, the set 𝜆∈ℂ∣min0,−𝑏₋ln|Im𝜆|<Re𝜆<max0,(2𝑛)⁻¹ln|Im𝜆|,|𝜆|>max𝑛⁴,𝜆_𝑛−1 (4.44) is open in ℂ, there exists such an 𝜀_𝑛>0 that this set, along with the point 𝜆_𝑛, contains the open disk of radius 𝜀_𝑛 centered at 𝜆_𝑛Δ_𝑛∶=𝜆∈ℂ∣𝜆−𝜆_𝑛<𝜀_𝑛. (4.45) Hence, for any 𝜆∈Δ_𝑛, 𝑛=1,2,…, min0,−𝑏₋ln|Im𝜆|<Re𝜆<max0,(2𝑛)⁻¹|,|𝜆|𝑛lnIm𝜆|>max⁴,𝜆_𝑛−1. (4.46) Since the points 𝜆_𝑛 are distinct, we can regard the radii of the disks, 𝜀_𝑛, to be small enough so that 0<𝜀_𝑛<1𝑛,𝑛=1,2,…,Δ_𝑖∩Δ_𝑗=∅,𝑖≠𝑗. (4.47)
By the properties of the s.m., the latter implies 𝐸_𝐴Δ_𝑖𝐸_𝐴Δ_𝑗=0,𝑖≠𝑗. (4.48) Observe that the subspaces 𝐸_𝐴(Δ_𝑛)𝑋, 𝑛=1,2,…, are nontrivial since Δ_𝑛∩𝜎(𝐴)≠∅, Δ_𝑛 being an open set.
By choosing a unit vector 𝑒_𝑛∈𝐸_𝐴(Δ_𝑛)𝑋 (‖𝑒_𝑛‖=1), 𝑛=1,2,…, we obtain a vector sequence such that 𝐸_𝐴Δ_𝑖𝑒_𝑗=𝛿_𝑖𝑗𝑒_𝑖. (4.49) In the same manner as in the proof of Theorem 4.1, one can show that there is an 𝜀>0 such that 𝑑_𝑛∶=dist𝑒_𝑛,span𝑒_𝑘∣𝑘=1,2,…,𝑘≠𝑛≥𝜀,𝑛=1,2,…. (4.50) As follows from the Hahn-Banach Theorem, for any 𝑛=1,2,…, there is an 𝑒^∗_𝑛∈𝑋^∗ such that 𝑒^∗_𝑛=1,𝑒_𝑖,𝑒^∗_𝑗=𝛿_𝑖𝑗𝑑_𝑖. (4.51) Concerning the sequence of the real parts, {Re𝜆_𝑛}^∞_𝑛=1, there are two possibilities: it is either bounded or unbounded.
First, assume that the sequence {Re𝜆_𝑛}^∞_𝑛=1 is bounded, that is, there is an 𝜔>0 such that Re𝜆_𝑛≤𝜔,𝑛=1,2,…. (4.52) Let 𝑓∶=^∞_𝑛=11𝑛²𝑒_𝑛. (4.53) By (4.49), 𝐸_𝐴∪^∞_𝑛=1Δ_𝑛𝑓=𝑓,𝐸_𝐴Δ_𝑛1𝑓=𝑛²𝑒_𝑛,𝑛=1,2,…. (4.54) In absolutely the same fashion as it was done in the case of bounded above sequence {Re𝜆_𝑛}^∞_𝑛=1 in the proof of the “only if” part of Theorem 4.1, it is shown that 𝑓∈⋂_0≤𝑡<∞𝐷(𝑒^𝑡𝐴).
Therefore (see Section 1), 𝑦(𝑡)∶=𝑒^𝑡𝐴𝑓, 0≤𝑡<∞, is a weak solution of (1.1) on [0,∞).
Let ℎ^∗∶=^∞_𝑛=11𝑛²𝑒^∗_𝑛∈𝑋^∗ (4.55) (cf. (4.51)).
Taking (4.50) and (4.51) into account, we have 𝑒_𝑛,ℎ^∗=𝑑_𝑛𝑛²≥𝜀𝑛²,𝑛=1,2,…. (4.56) Thus, _𝜎(𝐴)|𝜆|𝑒^Re𝜆𝑑𝑣𝑓,ℎ^∗()();=,𝜆considering4.54,by2.9_∞𝑛=11𝑛²_Δ𝑛|𝜆|𝑒^Re𝜆𝑑𝑣𝑒_𝑛,ℎ^∗,𝜆for𝜆∈Δ_𝑛,by(4.46),|𝜆|≥𝑛⁴;for𝜆∈Δ_𝑛,by(4.47)and(4.52),Re𝜆=Re𝜆_𝑛−Re𝜆_𝑛−Re𝜆≥Re𝜆_𝑛−Re𝜆_𝑛−Re𝜆≥−𝜔−𝜀_𝑛≥−𝜔−1;≥𝑒^{∞−(𝜔+1)}_𝑛=1𝑛²𝑣𝑒_𝑛,ℎ^∗,Δ_𝑛≥𝑒^{∞−(𝜔+1)}_𝑛=1𝑛²𝐸_𝐴Δ_𝑛𝑒_𝑛,ℎ^∗by(4.49)and(4.56);≥𝑒^{∞−(𝜔+1)}_𝑛=1𝑛²𝜀𝑛²=∞. (4.57) Whence, by the properties of the o.c. and [2], Proposition 3.1, 𝑦(1)=𝑒^𝐴𝑓∉𝐷(𝐴).
Consequently, by Proposition 3.1, the weak solution 𝑦(𝑡)=𝑒^𝑡𝐴𝑓,0≤𝑡<∞, of (1.1) on [0,∞) is not once strongly differentiable on (0,∞).
Now, assume that the sequence {Re𝜆_𝑛}^∞_𝑛=1 is unbounded. Therefore, there is a subsequence {Re𝜆_𝑛(𝑘)}^∞_𝑘=1 such that Re𝜆_𝑛(𝑘)⟶∞orRe𝜆_𝑛(𝑘)⟶−∞as𝑘⟶∞. (4.58) Suppose that Re𝜆_𝑛(𝑘)→∞ as 𝑘→∞.
Without restricting generality, we can regard that Re𝜆_𝑛(𝑘)≥𝑘,𝑘=1,2,…. (4.59) Considering this case just like the analogous one in the proof of the “only if” part of Theorem 4.1, one can show that the vector 𝑓∶=^∞_𝑘=1𝑒^{−𝑛(𝑘)Re𝜆𝑛(𝑘)}𝑒_𝑛(𝑘) (4.60) belongs to ⋂_0≤𝑡<∞𝐷(𝑒^𝑡𝐴) and, hence, (see Section 1) 𝑦(𝑡)∶=𝑒^𝑡𝐴𝑓, 0≤𝑡<∞, is a weak solution of (1.1) on [0,∞).
For any 𝜆∈Δ_𝑛(𝑘), 𝑘=1,2,…, by (4.46), (4.47), and (4.59), Re𝜆=Re𝜆_𝑛(𝑘)−Re𝜆_𝑛(𝑘)−Re𝜆≥Re𝜆_𝑛(𝑘)−Re𝜆_𝑛(𝑘)−Re𝜆≥Re𝜆_𝑛(𝑘)−𝜀_𝑛(𝑘)≥Re𝜆_𝑛(𝑘)−1𝑛(𝑘)((𝑘≥𝑘−1≥0,Re𝜆<max0,2𝑛))⁻¹ln|Im𝜆|, (4.61) Hence, for 𝜆∈Δ_𝑛(𝑘), 𝑘=1,2,…, |𝜆|≥|Im𝜆|≥𝑒^{2𝑛(𝑘)Re𝜆}≥𝑒^{2𝑛(𝑘)(Re𝜆𝑛(𝑘)−1/𝑛(𝑘))}. (4.62) Using this estimate, we have _𝜎(𝐴)|𝜆|𝑒^Re𝜆𝑑𝑣𝑓,ℎ^∗()();=,𝜆considering4.49,by2.9_∞𝑘=1𝑒^{−𝑛(𝑘)Re𝜆𝑛(𝑘)}_Δ𝑛(𝑘)|𝜆|𝑒^Re𝜆𝑑𝑣𝑒_𝑛(𝑘),ℎ^∗≥,𝜆_∞𝑘=1𝑒^{−𝑛(𝑘)Re𝜆𝑛(𝑘)}𝑒^{2𝑛(𝑘)(Re𝜆𝑛(𝑘)−1/𝑛(𝑘))}𝑣𝑒_𝑛(𝑘),ℎ^∗,Δ_𝑛(𝑘)=𝑒^∞−2_𝑘=1𝑒^{𝑛(𝑘)Re𝜆𝑛(𝑘)}𝐸_𝐴Δ_𝑛(𝑘)𝑒_𝑛(𝑘),ℎ^∗by(4.49),(4.56),and(4.59);≥𝑒⁻²𝜀^∞_𝑘=1𝑒^𝑛(𝑘)𝑛(𝑘)²=∞. (4.63) Whence, by the properties of the o.c. and [2], Proposition 3.1, 𝑦(1)=𝑒^𝐴𝑓∉𝐷(𝐴).
Therefore, by Proposition 3.1, the weak solution 𝑦(𝑡)=𝑒^𝑡𝐴𝑓, 0≤𝑡<∞, of (1.1) on [0,∞) is not once strongly differentiable on (0,∞).
Now, suppose that Re𝜆_𝑛(𝑘)→−∞ as 𝑘→∞.
Without restricting generality, we can regard that Re𝜆_𝑛(𝑘)≤−𝑘,𝑘=1,2,…. (4.64) Let 𝑓∶=^∞_𝑘=11𝑘²𝑒_𝑛(𝑘). (4.65) For any 𝑡≥0 and an arbitrary 𝑔^∗∈𝑋^∗, _𝜎(𝐴)𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗()();=,𝜆considering4.49,by2.9_∞𝑘=11𝑘^2Δ_𝑛(𝑘)𝑒^𝑡Re𝜆𝑑𝑣𝑒_𝑛(𝑘),𝑔^∗,𝜆for𝜆∈Δ_𝑛(𝑘),by(4.42)and(4.64),Re𝜆=Re𝜆_𝑛(𝑘)+Re𝜆−Re𝜆_𝑛(𝑘)≤Re𝜆_𝑛(𝑘)+𝜆−𝜆_𝑛(𝑘)≤≤−𝑘+1≤0;_∞𝑘=11𝑘²𝑣𝑒_𝑛(𝑘),𝑔^∗,Δ_𝑛(𝑘)();≤by2.8_∞𝑘=11𝑘²4𝑀𝑒_𝑛(𝑘)𝑔^∗=4𝑀𝑔^∗∞_𝑘=11𝑘²<∞. (4.66) Analogously, for an arbitrary 𝑡≥0, sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1(𝐴)𝜆∈𝜎∣𝑒𝑡Re𝜆>𝑛}𝑒^𝑡Re𝜆𝑑𝑣𝑓,𝑔^∗,𝜆≤sup_{𝑔∗∈𝑋∗∣‖𝑔∗}^∞_‖=1𝑘=11𝑘^2Δ_𝑛(𝑘)^{∩(𝐴)𝜆∈𝜎∣𝑒𝑡Re𝜆>𝑛}1𝑑𝑣𝑒_𝑛(𝑘),𝑔^∗,𝜆=sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1(𝐴)𝜆∈𝜎∣𝑒𝑡Re𝜆>𝑛}1𝑑𝑣𝑓,𝑔^∗,𝜆by(2.7);≤sup_{𝑔∗∈𝑋∗∣‖𝑔∗‖=1}4𝑀𝐸_𝐴𝜆∈𝜎(𝐴)∣𝑒^𝑡Re𝜆>𝑛𝑓𝑔^∗=4𝑀𝐸_𝐴𝜆∈𝜎(𝐴)∣𝑒^𝑡Re𝜆>𝑛𝑓bythestrongcontinuityofthe𝑠.𝑚.;⟶0as𝑛⟶∞. (4.67) From (4.66) and (4.67), by [2], Proposition 3.1, we infer that 𝑓∈⋂_0≤𝑡<∞𝐷(𝑒^𝑡𝐴).
Therefore (see Section 1), 𝑦(𝑡)∶=𝑒^𝑡𝐴𝑓, 0≤𝑡<∞, is a weak solution of (1.1) on [0,∞).
Let ℎ^∗∶=^∞_𝑘=11𝑘²𝑒^∗_𝑛(𝑘)∈𝑋^∗ (4.68) (cf. (4.51)).
Taking into account (4.50) and (4.51), we have 𝑒_𝑛(𝑘),ℎ^∗≥𝜀𝑘²,𝑘=1,2,…. (4.69) For any 𝜆∈Δ_𝑛(𝑘), 𝑘=1,2,…, by (4.47) and (4.64), Re𝜆=Re𝜆_𝑛(𝑘)+Re𝜆−Re𝜆_𝑛(𝑘)≤Re𝜆_𝑛(𝑘)+Re𝜆−Re𝜆_𝑛(𝑘)≤Re𝜆_𝑛(𝑘)+𝜀_𝑛(𝑘)≤Re𝜆_𝑛(𝑘)+1≤−𝑘+1≤0,Re𝜆>min0,−𝑏₋ln|Im𝜆|, (4.70) Hence, for 𝜆∈Δ_𝑛(𝑘), 𝑘=1,2,…, |𝜆|≥|Im𝜆|≥𝑒^{𝑏−−1[−Re𝜆]}. (4.71) Using these estimates, we have _𝜎(𝐴)|𝜆|𝑒^{(𝑏−−1/2)Re𝜆}𝑑𝑣𝑓,ℎ^∗()();=,𝜆considering4.49,by2.9_∞𝑘=11𝑘^2Δ_𝑛(𝑘)|𝜆|𝑒^{(𝑏−−1/2)Re𝜆}𝑑𝑣𝑒_𝑛(𝑘),ℎ^∗≥,𝜆_∞𝑘=11𝑘^2Δ_𝑛(𝑘)𝑒^{(𝑏−−1/2)[−Re𝜆]}𝑑𝑣𝑒_𝑛(𝑘),ℎ^∗,𝜆≥^∞_𝑘=1𝑒^{(𝑏−−1/2)[𝑘−1]}𝑘²𝑣𝑒_𝑛(𝑘),ℎ^∗,Δ_𝑛(𝑘)≥𝑒^{−𝑏−−1∞/2}_𝑘=1𝑒^{(𝑏−−1/2)𝑘}𝑘²𝐸_𝐴Δ_𝑛(𝑘)𝑒_𝑛(𝑘),ℎ^∗by(4.49),(4.64),and(4.69);≥𝑒^{−𝑏−−1/2}𝜀^∞_𝑘=1𝑒^{(𝑏−−1/2)𝑘}𝑘⁴=∞. (4.72) Whence, by the properties of the o.c. and [2], Proposition 3.1, 𝑦(𝑏₋/2)=𝑒^{(𝑏−−1/2)𝐴}𝑓∉𝐷(𝐴).
Therefore, by Proposition 3.1, the weak solution 𝑦(𝑡)=𝑒^𝑡𝐴𝑓, 0≤𝑡<∞, of (1.1) on [0,∞) is not once strongly differentiable on (0,∞).
With every possibility concerning {Re𝜆_𝑛}^∞_𝑛=1 considered, we infer that the opposite to the assumption that there is a 𝑏₊>0 such that, for any 𝑏₋>0, the set 𝜎(𝐴)⧵ℒ_{𝑏−,𝑏+} is bounded, allows to single out a weak solution of (1.1) on [0,∞) that is not once strongly differentiable on (0,∞), much less strongly infinite differentiable on (0,∞).
Thus, the “only if” part has been proved by contrapositive.

Theorem 5.2 of [3] and Theorem 4.1 of [4] have been generalized.

5. Certain Effects of Smoothness Improvement

As we observed in the proofs of the “only if” parts of Theorems 4.1 and 4.2, the opposits to the “if” parts’ premises imply that there is a weak solution of (1.1) on [0,∞), which is not strongly differentiable at 0 or, respectively, once strongly differentiable on (0,∞).

Therefore, the case of finite strong differentiability of the weak solutions is superfluous and we obtain the following effects of smoothness improvement.

Proposition 5.1. If every weak solution of (1.1) on [0,∞) is strongly differentiable at 0, then all of them are strongly infinite differentiable on [0,∞).

Proposition 5.2. If every weak solution of (1.1) on [0,∞) is once strongly differentiable on (0,∞), then all of them are strongly infinite differentiable on (0,∞).

These statements generalize Propositions 6.1 and 6.2 of [3], respectively, the latter agreeing with the case when 𝐴 is a linear operator (not necessarily spectral of scalar type) generating a 𝐶₀-semigroup (cf. [1, 13]).

6. Final Remarks

Due to the scalar type spectrality of the operator 𝐴, all the above criteria are formulated exclusively in terms of its spectrum, no restrictions on the operator's resolvent behavior necessary, which makes them inherently qualitative and more transparent than similar results for 𝐶₀-semigroups (cf. [14]).

If a scalar type spectral 𝐴 generates a 𝐶₀-semigroup (cf. [15]), we immediately obtain the results of paper [4] regarding infinite differentiability.

Acknowledgment

The author would like to express his gratitude to Mr. Michael and Jonell McCullough.