On the differentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis

Given the abstract evolution equation \[ y'(t)=Ay(t),\ t\in \mathbb{R}, \] with scalar type spectral operator $A$ in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly infinite differentiable on $\mathbb{R}$. The important case of the equation with a normal operator $A$ in a complex Hilbert space is obtained immediately as a particular case. Also, proved is the following inherent smoothness improvement effect explaining why the case of the strong finite differentiability of the weak solutions is superfluous: if every weak solution of the equation is strongly differentiable at $0$, then all of them are strongly infinite differentiable on $\mathbb{R}$.


Introduction
We find conditions on a scalar type spectral operator A in a complex Banach space 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 infinite differentiable on R. The important case of the equation with a normal operator A in a complex Hilbert space is obtained immediately as a particular case. We also prove the following inherent smoothness improvement effect explaining why the case of the strong finite differentiability of the weak solutions is superfluous: if every weak solution of the equation is strongly differentiable at 0, then all of them are strongly infinite differentiable on R.
The found results develop those of paper [15], where similar consideration is given to the strong differentiability of the weak solutions of the 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. [1]).
• Due to the closedness of A, a weak solution of equation (1.3) can be equivalently defined to be a strongly continuous vector function y : I → X such that, for all t ∈ I, t t0 y(s) ds ∈ D(A) and y(t) = y(t 0 ) + A t t0 y(s) ds, where t 0 is an arbitrary fixed point of the interval I, and is also called a mild solution (cf. [6, Ch. II, Definition 6.3], see also [17,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., [6,8]), i.e., the associated abstract Cauchy problem (ACP ) with f ∈ X (cf. [6, Ch. II, Proposition 6.4], see also [1,Theorem]), whereas the classical ones are those with f ∈ D(A) (see, e.g., [6, Ch. II, Proposition 6.3]).
• In our discourse, the associated ACP may be ill-posed, i.e., the scalar type spectral operator A need not generate a strongly continuous group of bounded linear operators (cf. [12]).

Preliminaries
Here, for the reader's convenience, we outline certain essential preliminaries.
Henceforth, unless specified otherwise, A is supposed to be 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 each Borel set δ of the complex plane C a projection operator E A (δ) on X and having the operator's spectrum σ(A) as its support [2,5].
Observe that, in a complex finite-dimensional space, the scalar type spectral operators are all linear operators on the space, for which there is an eigenbasis (see, e.g., [2,5]) and, in a complex Hilbert space, the scalar type spectral operators are precisely all those that are similar to the normal ones [20].
Associated with a scalar type spectral operator in a complex Banach space is the Borel operational calculus analogous to that for a normal operator in a complex Hilbert space [2,4,5,19], which assigns to any Borel measurable function F : σ(A) → C a scalar type spectral operator (see [2,5]).
In particular, . . } is the set of nonnegative integers, A 0 := I, I is the identity operator on X) and The properties of the spectral measure and operational calculus, exhaustively delineated in [2,5], underlie the entire subsequent discourse. Here, we underline a few facts of particular importance.
Due to its strong countable additivity, the spectral measure E A (·) is bounded [3,5], i.e., there is such an M > 0 that, for any Borel set δ ⊆ C, Observe that the notation · is used here to designate the norm in the space L(X) of all bounded linear operators on X. We adhere to this rather conventional economy of symbols in what follows also adopting the same notation for the norm in the dual space X * .
Further, for a Borel measurable function F : Indeed, since, for any Borel sets δ, σ ⊆ C, [2,5], for the total variation measure, Whence, due to the nonnegativity of F (·) (see, e.g., [7]), The following statement, allowing to characterize the domains of Borel measurable functions of a scalar type spectral operator in terms of positive 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 description of the weak solutions of equation (1.2) with a scalar type spectral operator A in a complex Banach space.  We also need the following characterizations of a particular weak solution's of equation (1.2) with a scalar type spectral operator A in a complex Banach space being strongly differentiable on a subinterval I of [0, ∞). is n times strongly differentiable on I iff y(t) ∈ D(A n ), t ∈ I, in which case, y (k) (t) = A k y(t), k = 1, . . . , n, t ∈ I.
Subsequently, the frequent terms "spectral measure" and "operational calculus" are abbreviated to s.m. and o.c., respectively.

General Weak Solution
Theorem 3.1 (General Weak Solution). Let A be a scalar type spectral operator in a complex Banach space (X, · ). A vector function y : R → X is a weak solution of equation (1.1) iff there is an f ∈ t∈R D(e tA ) such that (3.14) y(t) = e tA f, t ∈ R, the operator exponentials understood in the sense of the Borel operational calculus (see (2.8)).
Proof. As is noted in the Introduction, y : R → X is a weak solution of (1.1) iff Applying Theorem 2.1, to y + (·) and y − (·), we infer that, this is equivalent to the fact Remarks 3.1.
• More generally, Theorem 2.1 and its proof can be easily modified to describe in the same manner all weak solution of equation (1.3) for an arbitrary interval I of the real axis R.
• Theorem 3.1 implies, in particular, that the subspace t∈R D(e tA ) of all possible initial values of the weak solutions of equation (1.1) is the largest permissible for the exponential form given by (3.14), which highlights the naturalness of the notion of weak solution, and that associated ACP (1.5), whenever solvable, is solvable uniquely.
• Observe that the initial-value subspace t∈R D(e tA ) of equation (1.1), containing the dense in X subspace α>0 E A (∆ α )X, where which coincides with the class E {0} (A) of entire vectors of A of exponential type [16], is dense in X as well.
• When a scalar type spectral operator A in a complex Banach space generates a strongly continuous group {T (t)} t∈R of bounded linear operators, T (t) = e tA and D(e tA ) = X, t ∈ R, [12], and hence, Theorem 3.1 is consistent with the well-known description of the weak solutions for this setup (see (1.6)).
• Clearly, the initial-value subspace t∈R D(e tA ) of equation (1.1) is narrower than the initial-value subspace t≥0 D(e tA ) of equation (1.2) and the initial-value subspace t≥0 D(e t(−A) ) = t≤0 D(e tA ) of equation (1.4), in fact it is the intersection of the latter two.

Differentiability of a Particular Weak Solution
Here, we characterize a particular weak solution's of equation (1.1) with a scalar type spectral operator A in a complex Banach space being strongly differentiable on a subinterval I of R.
Proof. The statement immediately follows from the prior theorem and Proposition 2.2 applied to for an arbitrary weak solution y(·) of equation (1.1). 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 (1.1) is strongly infinite differentiable on I (y(·) ∈ C ∞ (I, X)) iff, for each t ∈ I, in which case y (n) (t) = A n y(t), n ∈ N, t ∈ I.

Infinite Differentiability of Weak Solutions
In this section, we characterize the strong infinite differentiability on R of all weak solutions of equation (1.1) with a scalar type spectral operator A in a complex Banach space.

Theorem 5.1 (Infinite Differentiability of Weak Solutions).
Let A be a scalar type spectral operator in a complex Banach space (X, · ) with spectral measure E A (·). Every weak solution of equation is bounded (see Fig. 1). By Theorem 3.1, Our purpose is to show that y(·) ∈ C ∞ (R, X), which, by Corollary 4.1, is attained by showing that, for each t ∈ R, Let us proceed by proving that, for any t ∈ R and m ∈ N For any t ∈ R, m ∈ N and an arbitrary g * ∈ X * , |λ| m e t Re λ dv(f, g * , λ) < ∞ 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 t ∈ R, m ∈ N and an arbitrary g * ∈ X * , Finally, for any t ∈ R, m ∈ N and an arbitrary g * ∈ X * , D(e tA ), by Proposition 2.1; Also, for any t ∈ R, m ∈ N and an arbitrary n ∈ N, Further, for any t ∈ R, m ∈ N and an arbitrary n ∈ N, as in (5.16); by the strong continuity of the s.m.; Finally, for any t ∈ R, m ∈ N and an arbitrary n ∈ N, |λ| m e t Re λ dv(f, g * , λ) as in (5.17); by the strong continuity of the s.m.; which further implies that, for each t ∈ R, Whence, by Corollary 4.1, we infer that which completes the proof of the "if" part.
"Only if" part. Let us prove this part by contrapositive assuming that, for any b + > 0 and b − > 0, the set σ(A) \ L b−,b+ is unbounded. In particular, this means that, for any n ∈ N, unbounded is the set Hence, we can choose a sequence of points {λ n } ∞ n=1 in the complex plane as follows: λ n ∈ σ(A), n ∈ N, − (2n) −1 ln | Im λ n | < Re λ n < (2n) −1 ln | Im λ n |, n ∈ N, The latter implies, in particular, that the points λ n , n ∈ N, are distinct (λ i = λ j , i = j).

(5.20)
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 {e n } ∞ n=1 in X such that (5.21) e n = 1, n ∈ N, and E A (∆ i )e j = δ ij e j , i, j ∈ N, where δ ij is the Kronecker delta.
As is easily seen, (5.21) implies that the vectors e n , n ∈ N, are linearly independent.
Furthermore, there is an ε > 0 such that Indeed, the opposite implies the existence of a subsequence d n(k) ∞ k=1 such that d n(k) → 0, k → ∞.
As follows from the Hahn-Banach Theorem, for any n ∈ N, there is an e * n ∈ X * such that (5.23) 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 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 (5.21)).
Let us consider separately each of the two cases.
Using this estimate, for the functional h * ∈ X * defined by (5.30), we have: With every possibility concerning {Re λ n } ∞ n=1 considered, we infer that assuming the opposite to the "if" part's premise allows to find a weak solution of (1.1) on [0, ∞) that is not strongly differentiable at 0, and hence, much less strongly infinite differentiable on R. with a scalar type spectral operator A in a complex Banach space can be strongly infinite differentiable while the operator A is unbounded, e.g., when A is an unbounded self-adjoint operator in a complex Hilbert space (cf. [10,Theorem 7.1]). This fact contrasts the situation when a closed densely defined linear operator A in a complex Banach space generates a strongly continuous group {T (t)} t∈R of bounded linear operators, i.e., the associated abstract Cauchy problem is well-posed (see Remarks 1.1), in which case even the (left or right) strong differentiability of all weak solutions of equation (1.1) at 0 immediately implies boundedness for A (cf. [6]).

The Cases of Normal and Self-Adjoint Operators
As an important particular case of Theorem 5.1, we obtain is bounded (see Fig. 1).
Remark 6.1. Corollary 6.1 develops the results of paper [10], where similar consideration is given to the strong differentiability of the weak solutions of equation (1.2) with a normal operator A in a complex Hilbert space on [0, ∞) and (0, ∞). Considering that, for a self-adjoint operator A in a complex Hilbert space X, σ(A) ⊆ R (see, e.g., [4,19]), we further arrive at

Inherent Smoothness Improvement Effect
As is observed in the proof of the "only if" part of Theorem 5.1, the opposite to the "if" part's premise implies that there is a weak solution of equation (1.1), which is not strongly differentiable at 0. This renders the case of finite strong differentiability of the weak solutions superfluous and we arrive at the following inherent effect of smoothness improvement.

Concluding Remark
Due to the scalar type spectrality of the operator A, Theorem 5.1 is stated exclusively in terms of the location of its spectrum in the complex plane, similarly to the celebrated Lyapunov stability theorem [9] (cf. [6, Ch. I, Theorem 2.10]), and thus, is an intrinsically qualitative statement (cf. [15,18]).

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

Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.