Some concrete operators and their properties

Abstract

We consider integration and double integration operators, the Hardy operator, and multiplication and composition operators on Lebesgue space Lp [0, 1] and Sobolev spaces W) , and we study their properties. In particular, we calculate norm and spectral multiplicity of the Hardy operator and some multiplication operators, investigate its extended eigenvectors, characterize some composition operators in terms of the extended eigenvectors of the Hardy operator, and calculate the numerical radius of the integration operator on the real L2 [0, 1] space. The main method for our investigation is the so-called Duhamel products method. Some other questions are also discussed and posed.

Some operator inequalities associated with Kantorovich and Hölder–McCarthy inequalities and their applications

“…So, it is natural to investigate Ber (A) and ber (A) for operators A ∈ B(H). For more detail informations about the Berezin set, Berezin number and their relations with the numerical range and numerical radius, the reader can be found in Karaev [14], Garayev et al [8], Garayev et al [9], Garayev et al [13], Gürdal et al [11], Gürdal et al [12], Yamancı et al [16], Altwaijry et al [10], and Garayev [7].…”

Gap Between Operator Norm and Spectral Radius for the Square of Antidiagonal Block Operator Matrices

“…In Banach space ordered by a normal and generating core, several inequalities for the spectral radius of a positive commutator of positive operators have been surveyed in [13]. The numerical range and numerical radius of some Volterra integral operator in Hilbert Lebesgue spaces at finite interval have been considered in [14,15].…”

Duhamel Banach algebra structure of some space and related topics

“…In general, the Duhamel product and α-Duhamel product has been extensively explored on various spaces by several authors in [8-10, 13, 24, 27, 34]. Many applications of Duhamel products have been well investigated, see, for example, [5,7,10,14,21,22,28,32]. In particular, Ivanova and Melikhov used Duhamel product in their recent works [15][16][17][18] in investigation of the commutant of Pommiez operator.…”