Covariant and Contravariant Symbols of Operators on $l^{2}(\mathbb{Z})$

Year 2020, Volume: 3 Issue: 2 , 116 - 124 , 15.12.2020

Abdelhamid S Elmabrok

https://doi.org/10.33401/fujma.718157

https://izlik.org/JA52AZ82HA

Abstract

In this paper, we investigate covariant and contravariant symbols of operators generated by a representation of the integer group $\mathbb{Z}$. Then we describe some properties (Existence, Uniqueness, Boundedness, Compactnessi and Finite rank) of these operators and reformulated some know results in terms of wavelet transform (covariant and contravariant symbols).

Keywords

Bounded , Compact and finite rank operators , Covariant and contravariant symbols of operators , Wavelet transformation

References

• [1] F. A. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1134–1167.
• [2] F. A. Berezin, Wick and anti-Wick symbols of operators, Mat. Sb. (N.S)., 86 (128) (1971), 578–610.
• [3] F. A. Berezine, Method of Second Quantization, Nauka, Moscow, 1988.
• [4] F. A. Berezin, General concept of quantization, Comm. Math. Phys., 40 (1975), 153–174.
• [5] V. V. Kisil, Calculus of operators: covariant transform and relative convolutions, Banach J. Math. Anal., 8 (2) (2014), 156–184.
• [6] V. V. Kisil, Wavelets in Banach spaces, Acta Appl. Math., 59 (1) (1999), 79–109.
• [7] A. V. Balakrishnan, Applied Functional Analysis, volume 3 of Applications of Mathematics, Springer-Verlag, New York, second edition, 1981.
• [8] P. A. Fillmore, J. P. Williams, On operator ranges, Advances in Math., 7 (1971), 254-218.
• [9] I. Gohberg, S. Goldberg, M. A. Kaashoek, Basic classes of linear operators, Birkh auser Verlag, Basel, 2003.
• [10] A. W. Naylor, G. R. Sell, Linear operator theory in engineering and science, volume 40 of Applied Mathematical Sciences, Springer-Verlag, New York, second edition, 1982.
• [11] H. Fuhr, Abstract Harmonic Analysis of Continuous Wavelet Transforms, Springer-Verlag Berlin Heidelberg, 2005.
• [12] L. Debnath, P. Mikusinski, Introduction to Hilbert Spaces with Applications, Academic Press, Boston, 2, 1999.
• [13] V. V. Kisil, Erlangen Programme at Large: An Overview, In S. V. Rogosin and A. A. Koroleva (eds.) Advances in applied analysis, Birkh¨auser Verlag, Basel, 2012, pp. 1–94.
• [14] V. V. Kisil, Integral representations and coherent states, Bull. Belg. Math. Soc. Simon Stevin., 2 (5) (1995), 529–540.
• [15] S. T. Ali, J. P. Antoine, J. P. Gazeau, Coherent States, Wavelets and Their Generalizations, Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 2000.
• [16] A. Perelomov, Generalized coherent states and their applications. Texts and Monographs in Physics, Springer-Verlag, Berlin, 1986.
• [17] F. A. Berezin, M. A. Shubin, The Schrodinger Equation, volume 66 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991.
• [18] V. V. Kisil, Covariant transforms, Journal of Physics: Conference Series., 284 (1) (2011), 12–38.
• [19] V. V. Kisil, Operator covariant transform and local principle, J. Phys. A: Math. Theor., 45 (2012), 1–10.
• [20] M. Garayev, S. Saltan, D. Gundogdu, On the inverse power inequality for the Berezin number of operators, Journal of Mathematical Inequalities., 12 (4) (2018), 997–1003.
• [21] J. R. Retherford, Hilbert Space: Compact Operators and the Trace Theorem, London Math. Soc. Monographs, Cambridge University Press Cambridge, 1993.
• [22] N. I. Akhiezer, I. M. Glazman, Theory of Operators in Hilbert Space, Pitman, Boston, I (1981), 254-218.
• [23] I. Chalendar, E. Fricain, M. G¨urdal, M. T. Karaev, Compactness and Berezin symbols, Acta Sci. Math., 78 (2012), 315-329.
• [24] M. T. Karaev, M. Gurdal, U. Yamancı, Special operator classes and their properties, Banach J. Math. Anal., 7 (2) (2013), 75-86.