Comment on “On the Carleman Classes of Vectors of a Scalar Type Spectral Operator”

Abstract

The results of three papers, in which the author inadvertently overlooks certain deficiencies in the descriptions of the Carleman classes of vectors, in particular the Gevrey classes, of a scalar type spectral operator in a complex Banach space established in “On the Carleman Classes of Vectors of a Scalar Type Spectral Operator,” Int. J. Math. Math. Sci. 2004 (2004), no. 60, 3219–3235, are observed to remain true due to more recent findings.

1. 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 [2–4] are observed not to affect the validity of the latter due to more recent findings of [5].

2. 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 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 a scalar type spectral operator defined as follows: where is the domain of an operator, is the characteristic function of a set , and is the set of natural numbers andare bounded scalar type spectral operators on defined in the same manner as for a normal operator (see, e.g., [9, 10]).

In particular, is the set of nonnegative integers, , and is the identity operator on andThe 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 , and the subspaceof infinite differentiable vectors of , the subspaces of , are called the Carleman classes of ultradifferentiable vectors of the operator corresponding to the sequence of Roumieu and Beurling type, respectively.

The inclusionsare obvious.

If two sequences of positive numbers and are related as follows:we also have the inclusionwith the sequences being subject to the conditiontheir corresponding Carleman classes coincide:In view of the latter, by Stirling’s formula, for , are the well-known Gevrey classes of ultradifferentiable vectors of of order of Roumieu and Beurling type, respectively (see, e.g., [11–13]). In particular, and are the classes of analytic and entire vectors of , respectively [14, 15], and and (i.e., the classes and corresponding to the sequence ) are the classes of entire vectors of of exponential and minimal exponential type, respectively (see, e.g., [13, 16]).

3. Remarks

If the sequence of positive numbers satisfies the conditionwhich can be equivalently stated as the scalar function first introduced by Mandelbrojt [17], is well defined (cf. [13]). The function is  continuous, strictly increasing, and .

As is shown in [11] (see also [12, 13]), the sequence satisfying the condition (WGR), for a normal operator in a complex Hilbert space , the equalities are true; the normal operatorsare understood in the sense of the Borel operational calculus (see, e.g., [9, 10]) and the function is replaceable with any nonnegative, continuous, and increasing on function satisfyingwith some and , in particular, with(cf. [13]).

Notably, when () with , the corresponding function is replaceable with (see [13] for details, cf. also [1]) andObserve that equalities (18) can be considered to be operator analogues of the classical Paley-Wiener Theorems relating the smoothness of the Fourier transform of a square-integrable on function to its decay at [18].

In [1, Theorem ] and [1, Corollary ], 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 inclusionsare proven only.

In the recent paper [5], the reflexivity requirement is shown to be superfluous and the following statements are proven.

Theorem 1 (see [5, Theorem ]). Let be a sequence of positive numbers satisfying the condition (WGR) given by (15). Then, for a scalar type spectral operator in a complex Banach space , equalities (18) are true; the scalar type spectral operators , , are understood in the sense of the Borel operational calculus and the function defined by (17) is replaceable with any nonnegative, continuous, and increasing on function satisfying condition (20).

Corollary 2 (see [5, Corollary ]). Let . Then, for a scalar type spectral operator in a complex Banach space , equalities (23) are true.

In papers [2–4], 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 ] and [4, Theorem ] and the sufficiency of [3, Theorem ]. Thus far, this circumstance leaving the statements effectively proven only for reflexive spaces, when, by [1, Theorem ] and [1, Corollary ], inclusions (24) and (25) turn into equalities (18) and (23), respectively, also seems to have escaped the attention of the referees and the authors who have cited [2, 3] (see, e.g., [19–21]).

However, the good news for all is that, due to [5, Theorem ] and [5, Corollary ], 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.

Conflicts of Interest

The author declares that there are no conflicts of interest.