Abstract
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.
Highlights
As was shown in [8], 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 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. 2
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
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
Summary
As was shown in [8] (see [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
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