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 Paley-Wiener type is considered as an application. 1. 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.


Introduction
Certain deficiencies of the descriptions (established in [1]) of the Carleman classes of vectors, in particular the Gevrey classes, of a scalar type spectral operator in a complex Banach space inadvertently overlooked by the author when proving the results of the three papers [2][3][4] are observed not to affect the validity of the latter due to more recent findings of [5].

Preliminaries
For the reader's convenience, we outline in this section certain preliminaries essential for understanding.
Henceforth, unless specified otherwise,  is supposed to be a scalar type spectral operator in a complex Banach space (, ‖ ⋅ ‖) and   (⋅) is supposed to be its strongly -additive spectral measure (the resolution of the identity) assigning to each Borel set  of the complex plane C a projection operator   () on  and having the operator's spectrum () as its support [6,7].
Observe that, in a complex finite-dimensional space, the scalar type spectral operators are those linear operators on the space, for which there is an eigenbasis (see, e.g., [6,7]) and, in a complex Hilbert space, the scalar type spectral operators are precisely those that are similar to the normal ones [8].
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 [6,7,9,10], which assigns to any Borel measurable function  : () → C a scalar type spectral operator defined as follows: where (⋅) is the domain of an operator,   (⋅) is the characteristic function of a set  ⊆ C, and N fl {1, 2, 3, . ..} is the set of natural numbers and are bounded scalar type spectral operators on  defined in the same manner as for a normal operator (see, e.g., [9,10]).
In particular, International Journal of Mathematics and Mathematical Sciences Z + fl {0, 1, 2, . ..} is the set of nonnegative integers,  0 fl , and  is the identity operator on  and The properties of the spectral measure and operational calculus are exhaustively delineated in [6,7].For a densely defined closed linear operator (, ()) in a (real or complex) Banach space (, ‖⋅‖), a sequence of positive numbers {  } ∞ =0 , and the subspace of infinite differentiable vectors of , the subspaces of  ∞ (), are called the Carleman classes of ultradifferentiable vectors of the operator  corresponding to the sequence {  } ∞ =0 of Roumieu and Beurling type, respectively.
The inclusions are obvious.
In [1, Theorem 3.1] and [1, Corollary 4.1], equalities ( 18) and ( 23) are generalized to the case of a scalar type spectral operator  in a reflexive complex Banach space , with the reflexivity requirement dropped, the inclusions ) , are proven only.
In the recent paper [5], the reflexivity requirement is shown to be superfluous and the following statements are proven.
In papers [2][3][4], written before [5], the deficiency of inclusions (24) and (25) for the general case is inadvertently overlooked by the author and wrong conclusions are drawn from them in the "only if " parts of [2, Theorem  23), respectively, also seems to have escaped the attention of the referees and the authors who have cited [2,3] (see, e.g., [19][20][21]).
However, the good news for all is that, due to [5, Theorem 3.1] and [5, Corollary 4.1], inclusions' (24) and (25) being actually equalities (18) and (23), respectively, without the requirement of reflexivity, readily amends the faulty logic in the proofs of all the foregoing statements, making them true for an arbitrary complex Banach space.