On normal $\mu$-Hankel operators

E. Yu. Kuzmenkova   ; A. R. Mirotin

Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 1, pp. 36-43

Résumé

Hankel operators form one of the most important classes of operators in spaces of analytic functions and have numerous implementations. These operators can be defined as operators having Hankel matrices (i. e., matrices whose elements depend only on the sum of the indices) with respect to some orthonormal basis in a separable Hilbert space. This work continues the research begun in the work of the authors «$\mu$-Hankel operators on Hilbert spaces», Opuscula Math., 2021, vol. 41, no. 6, p. 881–899, where a new class of operators in Hilbert spaces was introduced ($\mu$-Hankel operators, $\mu$ is a complex parameter). Such operators act in a separable Hilbert space and, in some orthonormal basis of this space, have matrices whose diagonals, orthogonal to the main diagonal, are geometric progressions with denominator $\mu$. Thus, the classical Hankel operators correspond to the case $\mu=1$. The main result of the paper is a criterion for the normality of $\mu$-Hankel operators. By analogy with the Hankel operators, the considered class of operators has specific implementations in the form of integral operators, which allows apply to these operators the results obtained in an abstract context, and thereby contribute to the theory of integral operators. In this paper, such a realization is considered in the Hardy space on the unit circle. Criteria for the self-adjointness and normality of these operators are given.

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