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

Authors: Marat V. Markin
Journal: International Journal of Mathematics and Mathematical Sciences
 (2018)

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## 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.


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## Body

## 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,A is supposed to be ascalar type spectral operator in a complex Banach space (X,·) and EA(·) is supposed to be its strongly σ-additivespectral measure (theresolution of the identity) assigning to eachBorel setδ of thecomplex planeC aprojection operatorEA(δ) on X and having the operator’sspectrumσ(A) as itssupport [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 aneigenbasis (see, e.g., [6, 7]) and, in a complex Hilbert space, the scalar type spectral operators are precisely those that are similar to thenormal ones [8].Associated with a scalar type spectral operator in a complex Banach space is theBorel operational calculus analogous to that for anormal operator in a complex Hilbert space [6, 7, 9, 10], which assigns to any Borel measurable function F:σ(A)→C a scalar type spectral operator (1)FA≔∫σAFλdEAλdefined as follows: (2)FAf≔limn→∞FnAf,f∈DFA,DFA≔f∈X∣limn→∞FnAfexists,where (3)Fn·≔F·χλ∈σAFλ≤n·,n∈N;D(·) is thedomain of an operator, χδ(·) is thecharacteristic function of a set δ⊆C, and N≔1,2,3,… is the set ofnatural numbers and(4)FnA≔∫σAFnλdEAλ,n∈N,arebounded scalar type spectral operators on X defined in the same manner as for anormal operator (see, e.g., [9, 10]).In particular,(5)An=∫σAλndEAλ,n∈Z+;Z+≔0,1,2,… is the set ofnonnegative integers, A0≔I, and I is theidentity operator on X and(6)ezA≔∫σAezλdEAλ,z∈C.The properties of thespectral measure andoperational calculus are exhaustively delineated in [6, 7].For a densely defined closed linear operator(A,D(A)) in a (real or complex) Banach space (X,·), a sequence of positive numbers mnn=0∞, and the subspace(7)C∞A≔⋂n=0∞DAnof infinite differentiable vectors of A, the subspaces of C∞(A), (8)CmnA≔f∈C∞A∣∃α>0∃c>0:Anf≤cαnmn,n∈Z+,CmnA≔f∈C∞A∣∀α>0∃c>0:Anf≤cαnmn,n∈Z+,are called theCarleman classes of ultradifferentiable vectors of the operator A corresponding to the sequence mnn=0∞ ofRoumieu andBeurling type, respectively.The inclusions(9)CmnA⊆CmnA⊆C∞Aare obvious.If two sequences of positive numbers{mn}n=0∞ and {mn′}n=0∞ are related as follows:(10)∀γ>0∃c=cγ>0:mn′≤cγnmn,n∈Z+,we also have the inclusion(11)Cmn′A⊆CmnA,with the sequences being subject to the condition(12)∃γ1,γ2>0,∃c1,c2>0:c1γ1nmn≤mn′≤c2γ2nmn,n∈Z+;their corresponding Carleman classes coincide:(13)CmnA=Cmn′A,CmnA=Cmn′A.In view of the latter, byStirling’s formula, for β≥0, (14)EβA≔Cn!βA=CnβnA,EβA≔Cn!βA=CnβnAare the well-knownGevrey classes of ultradifferentiable vectors of A of order β of Roumieu and Beurling type, respectively (see, e.g., [11–13]). In particular, E{1}(A) and E(1)(A) are the classes ofanalytic andentire vectors of A, respectively [14, 15], and E{0}(A) and E(0)(A) (i.e., the classes C{1}(A) and C(1)(A) corresponding to the sequence mn≡1) are the classes ofentire vectors of A ofexponential andminimal exponential type, respectively (see, e.g., [13, 16]).


## 3. Remarks

If the sequence of positive numbersmnn=0∞ satisfies the condition(15)WGR∀α>0∃c=cα>0:cαn≤mn,n∈Z+,which can be equivalently stated as (16)limn→∞mnn=∞,the scalar function (17)Tλ≔m0∑n=0∞λnmn,λ≥0,00≔1,first introduced by Mandelbrojt [17], is well defined (cf. [13]). The function is continuous,strictly increasing, and T(0)=1.As is shown in [11] (see also [12, 13]), the sequence mnn=0∞ satisfying the condition (WGR), for anormal operatorA in a complex Hilbert space X, the equalities (18)CmnA=⋃t>0DTtA,CmnA=⋂t>0DTtAare true; the normal operators(19)TtA≔∫σATtλdEAλ,t>0,are understood in the sense of the Borel operational calculus (see, e.g., [9, 10]) and the function T(·) is replaceable with anynonnegative,continuous, andincreasing on [0,∞) function F(·) satisfying(20)c1Fγ1λ≤Tλ≤c2Fγ2λ,λ≥R,with some γ1,γ2,c1,c2>0 and R≥0, in particular, with(21)Sλ≔m0supn≥0λnmn,λ≥0,orPλ≔m0∑n=0∞λ2nmn21/2,λ≥0.⁡(cf. [13]).Notably, whenmn≔[n!]β (mn≔nβn) with β>0, the corresponding function T(·) is replaceable with (22)Fλ=eλ1/β,λ≥0 (see [13] for details, cf. also [1]) and(23)EβA=⋃t>0DetA1/β,EβA=⋂t>0DetA1/β.Observe that equalities (18) can be considered to be operator analogues of the classicalPaley-Wiener Theorems relating the smoothness of theFourier transformf^(·) of a square-integrable on R function f(·) to its decay at ±∞ [18].In [1, Theorem 3.1] and [1, Corollary 4.1], equalities (18) and (23) are generalized to the case of ascalar type spectral operatorA in areflexive complex Banach space X, with the reflexivity requirement dropped, the inclusions(24)CmnA⊇⋃t>0DTtA,CmnA⊇⋂t>0DTtA,(25)EβA⊇⋃t>0DetA1/β,EβA⊇⋂t>0DetA1/βare 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 3.1]).
Let{mn}n=0∞ be a sequence of positive numbers satisfying the condition (WGR) given by (15). Then, for a scalar type spectral operator A in a complex Banach space (X,·), equalities (18) are true; the scalar type spectral operators T(tA), t>0, are understood in the sense of the Borel operational calculus and the function T(·) defined by (17) is replaceable with anynonnegative,continuous, andincreasing on [0,∞) function F(·) satisfying condition (20).Corollary 2 (see [5, Corollary 4.1]).
Letβ>0. Then, for a scalar type spectral operator A in a complex Banach space (X,·), 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 5.1] and [4, Theorem 5.1] and thesufficiency of [3, Theorem 3.1]. Thus far, this circumstance leaving the statements effectively proven only for reflexive spaces, when, by [1, Theorem 3.1] and [1, Corollary 4.1], 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 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.



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