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

Marat V Markin

doi:10.1155/2018/2135740

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.

Highlights

Certain deficiencies of the descriptions 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]
Unless specified otherwise, A is supposed to be a scalar type spectral operator in a complex Banach space (X, ‖ ⋅ ‖) and EA(⋅) is supposed to be its strongly σ-additive spectral measure assigning to each Borel set δ of the complex plane C a projection operator EA(δ) on X and having the operator’s spectrum σ(A) as its support [6, 7]
In a complex finite-dimensional space, the scalar type spectral operators are those linear operators on the space, for which there is an eigenbasis and, in a complex Hilbert space, the scalar type spectral operators are precisely those that are similar to the normal ones [8]

Summary

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

Remarks