ON THE CARLEMAN CLASSES OF VECTORS OF A SCALAR TYPE SPECTRAL OPERATOR

The Carleman classes of a scalar type spectral operator in a reflexive Banach space are characterized in terms of the operator’s resolution of the identity. A theorem of the PaleyWiener type is considered as an application.


Introduction.
As was shown in [8] (see also [9,10]), under certain conditions, the Carleman classes of vectors of a normal operator in a complex Hilbert space can be characterized in terms of the operator's spectral measure (the resolution of the identity).
The purpose of the present paper is to generalize this characterization to the case of a scalar type spectral operator in a complex reflexive Banach space.

The Carleman classes of vectors.
Let A be a linear operator in a Banach space X with norm • , {m n } ∞ n=0 a sequence of positive numbers, and (D(•) is the domain of an operator).The sets are called the Carleman classes of vectors of the operator A corresponding to the sequence {m n } ∞ n=0 of Roumie's and Beurling's types, respectively.Obviously, the inclusion holds.
The sequence {m n } ∞ n=0 will be subject to the following condition.(WGR) For any α > 0, there exist such a C = C(α) > 0 that Cα n ≤ m n , n= 0, 1, 2,.... (2.4)Note that the name WGR originates from the words "weak growth."Under this condition, the numerical function first introduced by Mandelbrojt [15], is well defined.This function is nonnegative, continuous, and increasing.As established in [8] (see also [9,10]), for a normal operator A with a spectral measure E A (•) in a complex Hilbert space H with inner product (•, •) and the sequence {m n } ∞ n=0 satisfying the condition (WGR), (2.6) the normal operators T (t|A|) (0 < t < ∞) being defined in the sense of the spectral operational calculus for a normal operator: where the function T (•) can be replaced by any nonnegative, continuous, and increasing with some positive γ 1 , γ 2 , c 1 , c 2 , and a nonnegative R.

Carleman ultradifferentiability.
Let I be an interval of the real axis, C ∞ (I) the set of all complex-valued functions strongly infinite differentiable on I, and {m n } ∞ n=0 a sequence of positive numbers.
Observe that Ᏹ {1} (I) is the class of the real analytic on I functions and Ᏹ (1) (I) is the class of entire functions, that is, the restrictions to I of analytic and entire functions, correspondingly, [15].
Note that condition (WGR), in particular, implies that lim n→∞ m n = ∞.Since, as is easily seen, the Carleman classes of vectors and functions coincide for the sequence {m n } ∞ n=1 and the sequence {dm n } ∞ n=1 for any d > 0, without loss of generality, we can regard that inf n≥0 m n ≥ 1. (2.12)

Scalar type spectral operators.
Henceforth, unless specified otherwise, A is a scalar type spectral operator in a complex Banach space X with norm • and E A (•) is its spectral measure (the resolution of the identity), the operator's spectrum σ (A) being the support for the latter [2,5].
Note that, in a Hilbert space, the scalar type spectral operators are those similar to the normal ones [21].
For such operators, there has been developed an operational calculus for Borel measurable functions on C (on σ (A)) [2,5], F(•) being such a function; a new scalar type spectral operator is defined as follows: is the domain of an operator), where (χ α (•) is the characteristic function of a set α), and being the integrals of bounded Borel measurable functions on σ (A), are bounded scalar type spectral operators on X defined in the same manner as for normal operators (see, e.g., [4,19]).The properties of the spectral measure, E A (•), and the operational calculus underlying the entire subsequent argument are exhaustively delineated in [2,5].We just observe here that, due to its strong countable additivity, the spectral measure E A (•) is bounded [3], that is, there is an M > 0 such that, for any Borel set δ, E A (δ) ≤ M. (2.17) Observe that, in (2.17), the notation • was used to designate the norm in the space of bounded linear operators on X.We will adhere to this rather common economy of symbols in what follows adopting the same notation for the norm in the dual space X * as well.Due to (2.17), for any f ∈ X and g * ∈ X * (X * is the dual space), the total variation v(f , g * , •) of the complex-valued measure E A (•)f , g * ( •, • is the pairing between the space X and its dual, X * ) is bounded.Indeed, δ being an arbitrary Borel subset of σ (A), [3], For the reader's convenience, we reformulate here [16, Proposition 3.1], heavily relied upon in what follows, which allows to characterize the domains of the Borel measurable functions of a scalar type spectral operator in terms of positive measures (see [16] for a complete proof).On account of compactness, the terms spectral measure and operational calculus for scalar type spectral operators, frequently referred to, will be abbreviated to s.m. and o.c., respectively.Proposition 2.1.Let A be a scalar type spectral operator in a complex Banach space X and F(•) a complex-valued Borel measurable function on C (on σ (A)).Then Observe that, for F(•) being an arbitrary Borel measurable function on C (on σ (A)), for any f ∈ D(F (A)), g * ∈ X * , and arbitrary Borel sets In particular, (2.22)

The Carleman classes of a scalar type spectral operator
Theorem 3.1.Let A be a scalar type spectral operator in a complex reflexive Banach space X.If a sequence of positive numbers {m n } ∞ n=0 satisfies condition (WGR), equalities (2.6) hold, the scalar type spectral operators T (t|A|) (0 < t < ∞) defined in the sense of the operational calculus for a scalar type spectral operator and the function T (•) being replaceable by any nonnegative, continuous, and increasing function with some positive γ 1 , γ 2 , c 1 , c 2 , and a nonnegative R.
On the other hand, (by the continuity of the s.m.) → 0 as n → ∞.  (3.10) and, for a certain (an arbitrary) α > 0, there is a c > 0 such that For any g * ∈ X * , (by the monotone convergence theorem) (3.12) Let By the properties of the o.c., T ((1/2α)|A|)E A (∆ n ), n = 0, 1, 2,..., is a bounded operator on X and (by the properties of the o.c.) (by the properties of the o.c.) → 0 as m → ∞. (3.15) Since a reflexive Banach space is weakly complete (see, e.g., [3]), we infer that the sequence weakly converges in X.This, considering the fact that, by the continuity of the s.m.,  Then, for a certain (any) t > 0, f ∈ D(T (t|A|)).
We infer from the latter that f ∈ C ∞ (A).Indeed, for an arbitrary N = 0, 1, 2,... and any g * ∈ X * , Remark 3.2.Observe that the assumption of the reflexivity of the space X was utilized for proving the inclusions The inverse inclusions hold regardless whether X is reflexive or not.
According to Stirling's formula, Hence, there is such a Taking this into account, we infer Now, we consider the family of functions It is easy to make sure that the function ρ λ (•) attains its maximum value on [0, ∞) at the point For λ ≥ e β , let N be the integer part of x λ = e −1 λ 1/β .Hence, N ≥ 1 and (4.7) Obviously, for all sufficiently large positive λ's, e −(βe −1 /2)λ Thus, by Theorem 3.1, in the considered case, the function T (λ) can be replaced by e λ 1/β (0 ≤ λ < ∞) and we arrive at the following.
Corollary 4.1.Let A be a scalar type spectral operator in a complex reflexive Banach space and 0 < β < ∞.Then In particular, for β = 1, Corollary 4.1 gives the description of the analytic and entire vectors of the scalar type spectral operator A.
Corollary 4.1 generalizes the corresponding result of [8] (see also [9,10]) for a normal operator in a complex Hilbert space.
Observe that the inclusions are valid without the assumption of the reflexivity of X (see Remark 3.2).

A theorem of the Paley-Wiener type. Consider the self-adjoint differential operator
With the unitary equivalence of this operator and the operator of multiplication by the independent variable x in view, by Theorem 3.1 as well as by [9,10], we arrive at the following theorem of the Paley-Wiener type [18,22].
The only natural question to be answered now is how the abstract smoothness relative to the differential operator A in L 2 (−∞, ∞) reveals itself as the smoothness in the ordinary sense.
For any f ∈ W n 2 (I), where I is an interval of the real axis and W n 2 (I) = H n (I) is the nth-order Sobolev space [20], let f (•) be the representative of the equivalence class f continuously differentiable n−1 times and such that f (n−1) (•) is absolutely continuous on I. For let f (•) be the infinite-differentiable representative of the equivalence class f such that We will impose upon the sequence {m n } ∞ n=0 an additional condition.(DI) There are an L > 0 and a γ > 1 such that m n+1 ≤ Lγ n m n , n= 0, 1, 2,....Note that the name (DI) originates from the words "differentiation invariant" since, as is easily verifiable, under this condition, the Carleman classes C {mn} (−∞, ∞) and C (mn) (−∞, ∞) along with a function f (•) contain its first derivative, f (•).

Remarks.
It is to be noted that, in [10] (see also [8,9]), not only were equalities (2.6) for a normal operator in a complex Hilbert space proved to hold in the set-theoretical sense but also in the topological sense, the sets C {mn} (A) and C (mn) (A) considered as the inductive and, respectively, projective limits of the Banach spaces and the sets t>0 D(T (t|A|)) and t>0 D(T (t|A|)) as the inductive and, respectively, projective limits of the Hilbert spaces with inner products Observe also that, in [11] (see also [10]), similar results were obtained for the generator of a bounded analytic semigroup in a Banach space.

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(3. 19 )
Now, we are to prove the inverse inclusions.