On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator of orders less than one

It is shown that, if all weak solutions of the evolution equation \begin{equation*} y'(t)=Ay(t),\ t\ge0, \end{equation*} with a scalar type spectral operator $A$ in a complex Banach space are Gevrey ultradifferentiable of orders less than one, then the operator $A$ is necessarily bounded.


Introduction
In [1][2][3], found are characterizations of the strong di erentiability and Gevrey ultradi erentiability of order β ≥ , in particular analyticity and entireness, on [ , ∞) and ( , ∞) of all weak solutions of the evolution equation with a scalar type spectral operator A in a complex Banach space. As is shown by [2,Theorem . ] (see also [2,Corollary . ]), all weak solutions of equation (1.1) can be entire vector functions, i.e., belong to the rst-order Beurling type Gevrey class E ( ) ([ , ∞), X) (see Preliminaries), while the operator A is unbounded, e.g., when A is a semibounded below self-adjoint operator in a complex Hilbert space (see [4,Corollary . ]). This remarkable fact contrasts the situation when, in (1.1), a closed densely de ned linear operator A generates a C -semigroup, in which case the strong di erentiability of all weak solutions of (1.1) at alone immediately implies boundedness for A (cf. [5], see also [6]).
It remains to examine whether all weak solutions of equation (1.1) with a scalar type spectral operator A in a complex Banach space can belong to the Gevrey classes of orders less than one (not necessarily to the same one) with A remaining unbounded.
In this paper, developing the results of [1][2][3], we show that an unbounded scalar type spectral operator A in a complex Banach space cannot sustain the strong Gevrey ultradi erentiability of all weak solutions of equation (1.1) for orders less than one, i.e., that imposing on all the weak solutions along with the entireness requirement certain growth at in nity conditions (see Preliminaries) necessarily makes the operator A bounded. Thus, we generalize the corresponding results for equation (1.1) with a normal operator A in a complex Hilbert space found in [7].
De nition 1.1 (Weak Solution). Let A be a densely de ned closed linear operator in a Banach space (X, · ). A strongly continuous vector function y : [ , ∞) → X is called a weak solution of equation (1.1) if, for any g * ∈ D(A * ), 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. [8] [9,Preliminaries]). -Such a notion of weak solution, which need not be di erentiable in the strong sense, generalizes that of classical one, strongly di erentiable on [ , ∞) and satisfying the equation in the traditional plug-in sense, the classical solutions being precisely the weak ones strongly di erentiable on [ , ∞). -When a closed densely de ned linear operator A in a complex Banach space X generates a C -semigroup T(t) t≥ of bounded linear operators (see, e.g., [5,10]), i.e., the associated abstract Cauchy problem (ACP) y (t) = Ay(t), t ≥ , is well-posed (cf. [5, Ch. II, De nition 6.8]), the weak solutions of equation (1.1) are the orbits with f ∈ X [5, Ch. II, Proposition 6.4] (see also [8,Theorem]), whereas the classical ones are those with f ∈ D(A) (see, e.g., [5, Ch. II, Proposition 6.3]). -In our discourse, the associated ACP need not be well-posed, i.e., the scalar type spectral operator A need not generate a C -semigroup (cf. [6]).

. Scalar Type Spectral Operators
Henceforth, unless speci ed 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 [11][12][13].
Observe that, in a complex nite-dimensional space, the scalar type spectral operators are all linear operators on the space, for which there exists an eigenbasis (see, e.g., [11][12][13]) and, in a complex Hilbert space, the scalar type spectral operators are precisely all those that are similar to the normal ones [14].
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 [12,13,15,16], which assigns to any Borel measurable function F : σ(A) → C a scalar type spectral operator are bounded scalar type spectral operators on X de ned in the same manner as for a normal operator (see, e.g., [15,16]). In particular, The properties of the spectral measure and operational calculus, exhaustively delineated in [12,13], 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 [13,17], i.e., there is such an M ≥ that, for any Borel set δ ⊆ C, (2.5) 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 * .
Also (Ibid.), for a Borel measurable function F : C → C, f ∈ D(F(A)), g * ∈ X * , and a Borel set δ ⊆ C, In particular, for δ = σ(A), Indeed, since, for any Borel sets δ, σ ⊆ C, Whence, due to the nonnegativity of F(·) (see, e.g., [20]), 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.

Proposition 2.1 ([21, Proposition . ]).
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. where The succeeding key theorem provides a description of the weak solutions of equation (1.1) with a scalar type spectral operator A in a complex Banach space.  (1.1) is the largest permissible for the exponential form given by (2.10), which highlights the naturalness of the notion of weak solution, and -that associated ACP (1.2), whenever solvable, is solvable uniquely.
-Observe that the initial-value subspace t≥ D(e tA ) of equation (1.1), containing the dense in X subspace -When a scalar type spectral operator A in a complex Banach space generates a C -semigroup T(t) t≥ , T(t) = e tA and D(e tA ) = X, t ≥ , [6], and hence, Theorem 2.1 is consistent with the well-known description of the weak solutions for this setup (see (1.3)).
Subsequently, the frequent terms "spectral measure" and "operational calculus" are abbreviated to s.m. and o.c., respectively.

. Gevrey Classes of Functions
De nition 2.1 (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 in nite di erentiable on an interval I ⊆ (−∞, ∞), and ≤ β < ∞.

. Gevrey Classes of Vectors
One can consider the Gevrey classes in a more general sense.
are called the βth-order Gevrey classes of ultradi erentiable vectors of A of Roumieu and Beurling type, respectively (see, e.g., [29][30][31]). -For ≤ β < β < ∞, the inclusions hold. -In particular, E { } (A) and E ( ) (A) are the classes of analytic and entire vectors of A, respectively [32,33] and E { } (A) and E ( ) (A) are the classes of entire vectors of A of exponential and minimal exponential type, respectively (see, e.g., [30,34]). -In view of the closedness of A, it is easily seen that the class E ( ) (A) forms the subspace of initial values in 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 ≤ β < being the subspaces of such initial values for which the solutions satisfy growth estimate (2.11) with some (any) γ > and some M > , respectively (cf. [28]).
In [19,35], descriptions (2.14) are extended to scalar type spectral operators in a complex Banach space, in which form they are basic for our discourse. In [35], similar nature descriptions of the classes E { } (A) and E ( ) (A) (β = ), known for a normal operator A in a complex Hilbert space (see, e.g., [30]), are also generalized to scalar type spectral operators in a complex Banach space. In particular [35,Theorem . ], 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 ultradi erentiable on a subinterval I of [ , ∞) proved in [9].

Proposition 2.2 ([9, Proposition . ]). Let A be a scalar type spectral operator in a complex Banach space (X, · ), ≤ β < ∞, and I be a subinterval of [ , ∞). Then the restriction of a weak solution y(·) of equation (1.1) to I belongs to the Gevrey class
in which case y (n) (t) = A n y(t), n ∈ N, t ∈ I.

One Lemma
The following lemma generalizes [7, Lemma . ], its counterpart for a normal operator in a complex Hilbert space and, besides being an interesting result by itself, is necessary for proving our main statement.

Lemma 3.1. Let A be a scalar type spectral operator in a complex
Banach space (X, · ) with spectral measure E A (·) and < β < ∞. If then the operator A is bounded.

Proof.
First, observe that, in view of inclusions (2.12), for any β > , Let us prove the statement by contrapositive assuming A to be unbounded. The operator A being scalar type spectral, this assumption implies that the spectrum σ(A) of A is an unbounded set in the complex plane C [12,13]. Hence, the points of the spectrum can be found in in nitely many semi-open annuli of the form i.e., there is a sequence of natural numbers {n(k)} ∞ k= such that k ≤ n(k) < n(k + ), k ∈ N, (3.16) and, for each k ∈ N, there is a λ k ∈ δ n(k) ∩ σ(A) ≠ ∅.
Observe also, that the subspaces E A (∆ k )X, k ∈ N, are nontrivial since with ∆ k being an open set in C.
By choosing a unit vector e k ∈ E A (∆ k )X for each k ∈ N, we obtain a sequence {en} ∞ n= in X such that where δ ij is the Kronecker delta.
As is easily seen, (3.19) implies that the vectors e k , k ∈ N, are linearly independent. Furthermore, there is an ε > such that Then, by selecting a vector we arrive at = e k(m) since, by (3.19), which is a contradiction proving (3.20).
As follows from the Hahn-Banach Theorem, for any k ∈ N, there is an e * k ∈ X * such that e * k = , k ∈ N, and e i , e * j = δ ij d i , i, j ∈ N. the elements being well de ned since e k = , k ∈ N (see (3.19)) and (l is the space of absolutely summable sequences). Indeed, in view of (3.16) and (3.17), for all k ∈ N su ciently large so that we have: In view of (3.19), by the properties of the s.m., and For an arbitrary t > and any g * ∈ X * , Indeed, in view of (3.16) and (3.17), for all k ∈ N su ciently large so that we have: Similarly, for any t > and n ∈ N, since, by (3.16) and (3.17), there is an L > such that the functional being well de ned since, by (3.16), {n(k) − } ∞ k= ∈ l and e * k = , k ∈ N (see (3.21)). In view of (3.21) and (3.20), we have: Fixing an arbitrary < β < β , for any t > , we have: since, by (3.18) and (3.17), for λ ∈ ∆ k , |λ| > n(k) − ε k > ; with some < β < (β depends on y(·)). This, by Proposition 2.2, implies that In particular, Since the equlity here being necessary and su cient for −A to be a generator of an analytic C -semigroup [36] (see also [5,10]). Observe that inclusion (4.33) also directly follows from the fact that E ( ) (A) is the subspace of initial values in X generating the (classical) solutions of (1.1), which are entire vector functions represented by power series (2.13) (see Preliminaries).