On Normal \(\mu\)-Hankel Operators

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/t8778-6480-0136-d On Normal \(\mu\)-Hankel Operators Kuzmenkova, E. Yu. , Mirotin, A. R. Vladikavkaz Mathematical Journal 2022. Vol. 24. Issue 1. Abstract: 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. Keywords: Hankel operator, \(\mu\)-Hankel operator, normal operator, self-adjoint operator, Hardy space, integral operator Language: Russian Download the full text For citation: Kuzmenkova, E. Yu. and Mirotin, A. R. On Normal \(\mu\)-Hankel Operators, Vladikavkaz Math. J., 2021, vol. 24, no. 1, pp. 36-43 (in Russian). DOI 10.46698/t8778-6480-0136-d + References 1. Nikolski, N. K. Operators, Functions, and Systems: an Easy Reading. Vol. 1: Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs, Vol. 92, American Mathematical Society, 2002, 461 p. 2. Nikolski, N. K. Operators, Functions, and Systems: an Easy Reading. Vol. 2: Model Operators and Systems, Mathematical Surveys and Monographs, Vol. 93, American Mathematical Society, 2002, 438 p. 3. Peller, V. V. Hankel Operators and Their Applications, Springer Monographs in Mathematics, Berlin, Springer-Verlag, 2003, 784 p. 4. Ho, M. C. On the Rotational Invariance for the Essential Spectrum of \(\lambda\)-Toeplitz Operators, J. Math. Anal. Appl., 2014, vol. 413, no. 2, pp. 557-565. DOI: 10.1016/j.jmaa.2013.11.056. 5. Mirotin, A. R. On the Essential Spectrum of \(\lambda\)-Toeplitz Operators over Compact Abelian Groups, J. Math. Anal. Appl., 2015, vol. 424, pp. 1286-1295. 6. Mirotin, A. R. and Kuzmenkova, E. Yu. \(\mu\)-Hankel Operators on Hilbert Spaces, Opuscula Math., 2021, vol. 41, no. 6, pp. 881-899. 7. Martнnez-Avendano, R. A. and Rosenthal, P. An Introduction to Operators on the Hardy-Hilbert Space, New York, Springer, 2007, 229 p. 8. Akhiezer, N. I. and Glazman, I. M. Theory of Linear Operators in Hilbert Space, Translated from the Russian by Merlynd Nestell, New York, Dover Publications, 661 p. 9. Mirotin, A. R. and Kovalyova, I. S. The Markov-Stieltjes Transform on Hardy and Lebesgue Spaces, Integral Transforms and Special Functions, 2016, vol. 27, no. 12, pp. 995-1007. DOI: 10.1080/10652469.2016.1247074. Corrigendum to our paper ``The Markov-Stieltjes transform on Hardy and Lebesgue spaces'', Integral Transforms and Special Functions, 2017, vol. 28, no. 5, pp. 421-422. DOI: 10.1080/10652469.2017.1298592. 10. Mirotin, A. R. and Kovalyova, I. S. Generalized Markov-Stieltjes Operator on Hardy and Lebesgue Spaces, Tr. Inst. Mat., 2017, vol. 25, no. 1, pp. 39-50. 11. Weidmann, J. Linear Operators in Hilbert Spaces, New York, Springer, 1980, 415 p. 12. Zhu, K. Operator Theory in Function Spaces, Second Edition, Mathematical Surveys and Monographs, vol. 138, American Mathematical Society, 2007, 348 p. 13. Prudnikov, A. P., Brychkov, Yu. A. and Marichev, A. I. Integrals and series: Elementary functions, New York, Gordon and Breach, 1986, 808 p. 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