On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis

Found are conditions on a scalar type spectral operator $A$ in a complex Banach space necessary and sufficient for all \textit{weak solutions} of the evolution equation \begin{equation*} y'(t)=Ay(t),\ t\ge 0, \end{equation*} to be strongly Gevrey ultradifferentiable of order $\beta\ge 1$, in particular analytic, on the open semi-axis $(0,\infty)$. A certain inherent smoothness improvement effect is analyzed.


Introduction
We find conditions on a scalar type spectral operator A in a complex Banach space, formulated exclusively in terms of the location of its spectrum in the complex plane, necessary and sufficient for all weak solutions of the evolution equation (1.1) y (t) = Ay(t), t ∈ R, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order β ≥ 1, in particular analytic or entire, on R. We also reveal certain inherent smoothness improvement effects and show that, if all weak solutions of the equation are Gevrey ultradifferentiable of orders less than one, then the operator A is necessarily bounded.
The important particular case of the equation with a normal operator A in a complex Hilbert follows immediately.
The found results develop those of paper [29] on the strong differentiability of the weak solutions of equation (1.1) on R and of papers [30][31][32], where similar consideration is given to the Gevrey ultradifferentiability of the weak solutions of the evolution equation where 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 * (cf. [2]).
• Due to the closedness of A, a weak solution of equation (2.3) can be equivalently defined to be a strongly continuous vector function y : I → X such that where t 0 is an arbitrary fixed point of the interval I, and is also called a mild solution (cf. [9, Ch. II, Definition 6.3], see also [28,Preliminaries]).
• Such a notion of weak solution, which need not be differentiable in the strong sense, generalizes that of classical one, strongly differentiable on I and satisfying the equation in the traditional plug-in sense, the classical solutions being precisely the weak ones strongly differentiable on I.
• As is easily seen y : R → X is a weak solution of equation ( • When a closed densely defined linear operator A in a complex Banach space X generates a strongly continuous group {T (t)} t∈R of bounded linear operators (see, e.g., [9,15]), i.e., the associated abstract Cauchy problem (ACP ) with f ∈ X (cf. [9, Ch. II, Proposition 6.4], see also [2,Theorem]), whereas the classical ones are those with f ∈ D(A) (see, e.g., [9, Ch. II, Proposition 6.3]).
Henceforth, unless specified otherwise, A is a scalar type spectral operator in a complex Banach space (X, · ) with strongly σ-additive spectral measure (the resolution of the identity) E A (·) assigning to Borel sets of the complex plane C projection operators on X and having the operator's spectrum σ(A) as its support [4,5,8].
Observe that, in a complex Hilbert space, the scalar type spectral operators are precisely all those that are similar to the normal ones [7,36,38].
Associated with a scalar type spectral operator A is the Borel operational calculus assigning to complex-valued Borel measurable functions defined on σ(A) scalar type spectral operators [8].
Provided σ(A) ⊆ {λ ∈ C | Re λ ≤ ω} with some ω ∈ R, the collection of exponentials e tA t≥0 is the C 0 -semigroup generated by A [24, Proposition 3.1] (cf. also [3,35]), and hence, if with some ω ≥ 0, the collection of exponentials e tA t∈R is the strongly continuous group of bounded linear operators generated by A.
Remark 2.2. The notation · is used here to designate the norm on the space L(X) of all bounded linear operators on X. Henceforth, we adhere to this rather conventional economy of symbols adopting the same notation also for the norm on the dual space X * .
The following statement characterizing the domains of Borel measurable functions of a scalar type spectral operator in terms of Borel measures is fundamental for our discourse.
Let A be a scalar type spectral operator in a complex Banach space (X, · ) with spectral measure E A (·) and F : σ(A) → C be a Borel measurable function. Then where v(f, g * , ·) is the total variation measure of E A (·)f, g * .
The succeeding key theorem provides a full description of the weak solutions of equation (1.1) with a scalar type spectral operator A in a complex Banach space.
the operator exponentials understood in the sense of the Borel operational calculus (see (2.7)).
We also need the following characterization of a particular weak solution's of equation (1.1) with a scalar type spectral operator A in a complex Banach space being strongly infinite differentiable on a subinterval I of R.

Proposition 2.2 ([29, Corollary 11]).
Let A be a scalar type spectral operator in a complex Banach space (X, · ) and I be a subinterval of R. A weak solution y(·) of equation in which case y (n) (t) = A n y(t), n ∈ N, t ∈ I.

Gevrey Classes of Functions.
Definition 2.2 (Gevrey Classes of Functions). Let (X, · ) be a (real or complex) Banach space, C ∞ (I, X) be the space of all X-valued functions strongly infinite differentiable on an interval I ⊆ R, and 0 ≤ β < ∞.
• For β = 1, E {1} (I, X) is the class of all analytic on I, i.e., analytically continuable into complex neighborhoods of I, vector functions and E (1) (I, X) is the class of all entire, i.e., allowing entire continuations, vector functions [20].

Definition 2.3 (Gevrey Classes of Vectors).
Let A be a densely defined closed linear operator in a (real or complex) Banach space (X, · ) and 0 ≤ β < ∞.
• In view of Stirling's formula, the sequence [n!] β ∞ n=0 can be replaced with hold.
• In particular, E {1} (A) and E (1) (A) are the classes of analytic and entire vectors of A, respectively [11,34] and E {0} (A) and E (0) (A) are the classes of entire vectors of A of exponential and minimal exponential type, respectively (see, e.g., [14,37]).
• In view of the closedness of A, it is easily seen that the class E (1) (A) forms the subspace of the initial values f ∈ X generating the (classical) solutions of (1.1), which are entire vector functions represented by the power series the classes E {β} (A) and E (β) (A) with 0 ≤ β < 1 being the subspaces of such initial values for which the solutions satisfy growth estimate (2.14) with some (any) γ > 0 and some M > 0, respectively (cf. [19]).
In [26,27], descriptions (2.15) are extended to scalar type spectral operators in a complex Banach space, in which form they are basic for our discourse. In [27], similar nature descriptions of the classes E {0} (A) and E (0) (A) (β = 0), known for a normal operator A in a complex Hilbert space (see, e.g., [14]), are also generalized to scalar type spectral operators in a complex Banach space. In particular [27, 2.5. Gevrey Ultradifferentiability of a Particular Weak Solution of (1.2).
We also need the following characterization of a particular weak solution's of equation (1.1) with a scalar type spectral operator A in a complex Banach space being strongly Gevrey ultradifferentiable on a subinterval I of [0, ∞).

Gevrey Ultradifferentiability of a Particular Weak Solution
Here, we characterize the strong Gevrey ultradifferentiability of a particular weak solution of equation (1.1) with a scalar type spectral operator A in a complex Banach space on an interval of the real axis.

Proposition 3.1 (Gevrey Ultradifferentiability of a Particular Weak Solution).
Let A be a scalar type spectral operator in a complex Banach space (X, · ), 0 ≤ β < ∞, and I be an interval of the real axis R. The restriction of a weak solution y(·) of equation (1.1) to I belongs to the Gevrey class Proof. As is noted in Remarks 2.1, y : R → X is a weak solution of (1.1) iff is a weak solution of equation (1.2) and The statement immediately follows from Proposition 2.3 applied to for an arbitrary weak solution y(·) of equation (1.1).

Gevrey Ultradifferentiability of Weak Solutions
In this section, we characterize the strong Gevrey ultradifferentiability of order β ≥ 1 on R of all weak solutions of equation (1.1) with a scalar type spectral operator A in a complex Banach space.
Theorem 4.1. Let A be a scalar type spectral operator in a complex Banach space (X, · ) with spectral measure E A (·) and 1 ≤ β < ∞. Then the following statements are equivalent.
Proof. We are to prove the closed chain of implications the implication (i) ⇒ (ii) following immediately from the inclusion To prove the implication (iii) ⇒ (i), suppose that there exist b + > 0 and b − > 0 such that the set σ(A)\P β b−,b+ is bounded and let y(·) be an arbitrary weak solution of equation (1.1).
By Theorem 2.1, Our purpose is to show that y(·) ∈ E (β) (R, X), which, by Proposition 3.1 and (2.15), is accomplished by showing that, for each t ∈ R, Let us proceed by proving that, for any t ∈ R and s > 0, For any s > 0, t ∈ R and an arbitrary g * ∈ X * , due to the boundedness of the sets , the continuity of the integrated function on C, and the finiteness of the measure v(f, g * , ·).
Further, for any s > 0, t ∈ R and an arbitrary g * ∈ X * , since, in view of Re λ ≥ 1 and β ≥ 1, Re λ β ≥ Re λ; Finally, for any s > 0, t ∈ R and an arbitrary g * ∈ X * , Also, for any s > 0, t ∈ R and an arbitrary n ∈ N, Indeed, since, due to the boundedness of the sets and the continuity of the integrated function on C, the sets are empty for all sufficiently large n ∈ N, we immediately infer that, for any s > 0 and t ∈ R, Further, for any s > 0, t ∈ R and an arbitrary n ∈ N, as in (4.17); by the strong continuity of the s.m.; Finally, for any s > 0, t ∈ R and an arbitrary n ∈ N, by the strong continuity of the s.m.; which, in view of (2.15), further implies that, for each t ∈ R, Whence, by Proposition 3.1, we infer that which completes the proof of the implication (iii) ⇒ (i).
Furthermore, we can regard the radii of the disks to be small enough so that 0 < ε n < 1 n , n ∈ N, and ∆ i ∩ ∆ j = ∅, i = j (i.e., the disks are pairwise disjoint).

(4.21)
Whence, by the properties of the s.m., where 0 stands for the zero operator on X.
Observe also, that the subspaces E A (∆ n )X, n ∈ N, are nontrivial since with ∆ n being an open set in C.
By choosing a unit vector e n ∈ E A (∆ n )X for each n ∈ N, we obtain a sequence where δ ij is the Kronecker delta.
As is easily seen, (4.22) implies that the vectors e n , n ∈ N, are linearly independent.
Furthermore, there exists an ε > 0 such that Indeed, the opposite implies the existence of a subsequence d n(k) Then, by selecting a vector such that e n(k) − f n(k) < d n(k) + 1/k, k ∈ N, we arrive at 1 = e n(k) since, by (4.22), E A (∆ n(k) )f n(k) = 0; = E A (∆ n(k) )(e n(k) − f n(k) ) ≤ E A (∆ n(k) ) e n(k) − f n(k) by (2.10); which is a contradiction proving (4.23).
As follows from the Hahn-Banach Theorem, for any n ∈ N, there is an e * n ∈ X * such that (4.24) e * n = 1, n ∈ N, and e i , e * j = δ ij d i , i, j ∈ N.
Let us consider separately the two possibilities concerning the sequence of the real parts {Re λ n } ∞ n=1 : its being bounded or unbounded. First, suppose that the sequence {Re λ n } ∞ n=1 is bounded, i.e., there is such an ω > 0 that (4.25) | Re λ n | ≤ ω, n ∈ N, and consider the element which is well defined since k −2 ∞ k=1 ∈ l 1 (l 1 is the space of absolutely summable sequences) and e k = 1, k ∈ N (see (4.22)).
In view of (4.24) and (4.23), we have: For any s > 0, which, by Proposition 3.1, implies that the weak solution y(t) = e tA f , t ∈ R, of equation (1.1) does not belong to the Roumieu type Gevrey class E {β} (R, X) and completes our consideration of the case of the sequence's {Re λ n } ∞ n=1 being bounded. Now, suppose that the sequence {Re λ n } ∞ n=1 is unbounded. Therefore, there is a subsequence {Re λ n(k) } ∞ k=1 such that Re λ n(k) → ∞ or Re λ n(k) → −∞, k → ∞.
Let us consider separately each of the two cases.

First, suppose that
Re λ n(k) → ∞, k → ∞ Then, without loss of generality, we can regard that Consider the elements and e n(k) = 1, k ∈ N (see (4.22)).
The remaining case of Re λ n(k) → −∞, k → ∞ is symmetric to the case of Re λ n(k) → ∞, k → ∞ and is considered in absolutely the same manner, which furnishes a weak solution y(·) of equation (1.1) such that and hence, by Proposition 3.1, not belonging to the Roumieu type Gevrey class E {β} (R, X).
With every possibility concerning {Re λ n } ∞ n=1 considered, the proof by contrapositive of the "only if" part is complete and so is the proof of the entire statement.
Remark 4.1. Due to the scalar type spectrality of the operator A, Theorem 4.1 is stated exclusively in terms of the location of its spectrum in the complex plane, and hence, is an intrinsically qualitative statement (cf. [29,30,32]).  For β = 1, we obtain the following important particular case.  Figure 2. The case of β = 1.

Remark 4.2.
As is established in [29], all weak solutions of equation ( Notably, for β = 1, we have: if every weak solution of equation (1.1) with a scalar type spectral operator A in a complex Banach space X is analytically continuable into a complex neighborhood of R (each one into its own), then all of them are entire vector functions, which can be further strengthened as follows. Let a weak solution y(·) of equation (1.1) be analytically continuable into a complex neighborhood of 0. This implies that there is a δ > 0 such that The power series converging at t = δ, there is a c > 0 such that Whence, considering that, by Proposition 3.1 with I = [−δ, δ], y(0) ∈ C ∞ (A) and y (n) (0) = A n y(0), n ∈ Z + , we infer that A n y(0) = y (n) (0) ≤ c δ −1 n n!, n ∈ Z + , which implies Now, let us prove the statement by contrapositive assuming that there is a weak solution of equation (1.1), which is not an entire vector function. This, by Theorem 4.1 with β = 1, implies that there is a weak solution y(·) of equation (1.1), which is not analytically continuable into a complex neighborhood of R. Then, by Proposition 3.1, for some t 0 ∈ R, Therefore, for the weak solution which, as is shown above, implies that y t0 (·) is not analytically continuable into a complex neighborhood of 0, and hence, completes the proof by contrapositive.
Similarly, considering the scalar type spectral operator z n n! A n f, z ∈ C, with some f ∈ X, and hence, satisfying the growth condition y(z) ≤ f e A |z| , z ∈ C, is of exponential type (see Preliminaries).
From Corollary 4.1, we immediately obtain the following Considering that, for a self-adjoint operator A in a complex Hilbert space X, σ(A) ⊆ R (see, e.g., [7,36]), by Corollary 4.2, we can strengthen [29,Corollary 18] as follows. For a normal operator in a complex Hilbert space, Proposition 5.1 and Theorem 6.1 acquire the following forms, respectively.

Acknowledgments
The author's appreciation goes to his colleague, Dr. Maria Nogin of the Department of Mathematics, California State University, Fresno, for her kind assistance with the graphics.