On the spectrum and Hilbert Schimidt properties of generalized Rhaly matrices

Abstract

In this paper, we show that the multiplications of some generalized Rhaly operators are Hilbert Schimidt operators.We also calculate the spectrum and essential spectrum of a special generalized Rhaly matrices on the Hardy space H².

Keywords

• Cesàro operators,terraced operators,Rhaly operators,generalized Cesàro operators,generalized terraced operators,spectrum,essential spectrum,Hilbert Schimidt

References

1. Akhmedov, A. M. and Başar, F., The fine spectra of the Cesaro operator C₁ over the sequence space bv_{p}, Math. J. Okayama Univ., 50 (2007), 135-147.
2. Aleman, A. and Siskakis, A. G., An integral Operator on H^{p}, Complex Variables, 28 (1995), 149-158.
3. Brown, A., Halmos, P. R. and Shields, A. L., Cesàro Operators, Acta Sci. Math., 26 (1965), 125-134.
4. Cima, J. A. and Petersen, K .E., Some Analytic Functions Whose Boundary Values Have Bounded Mean Oscillation, Math. Z., 147 (1976), 237-247.
5. Conway, J. B., A Course in Operator Theory, AMS, 2000.
6. Durna, N. and Yildirim, M., Topological resuls about the set of generalized Rhally matrices, Gen. Math. Notes, 17(1) (2013), 1-7.
7. Durna, N. and Yildirim, M., Generalized terraced matrices, Miskolc Math. Notes, 17(1) (2016), 201-208.
8. Garnett, J. B., Bounded analytic functions, vol. 96 of Pure and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981.

9. Gonzalez, M., The fine spectrum of the Cesàro operator in ℓ_{p} (1
10. Hardy, G. H., Littlewood, J. E. and Polya, G., Inequalities, Cambridge, 1934.
11. Okutoyi, I. J., On the spectrum of C₁ as an operator on bv₀, J. Austral. Math. Soc. Series A, 48 (1990), 79-86.
12. Pommerenke, C., Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation, Comment. Math. Helv., 52 (1977), 591-602.
13. Rhaly, Jr. H. C., Terraced matrices, Bull. London Math. Soc., 21 (1989), 399-406.
14. Rhoades, B. E. and Yildirim, M., The spectra of factorable matrices on ℓ_{p}, Integr. Equ. Oper. Theory, 55 (2006), 111-126.
15. Siskakis, A. G., Composition Semigroups and the Cesàro Operator on H^{p}, J. London Math. Soc., 36 (1987), 153-164.
16. Valeriyevna, G. O., Strength Properties of Safara Essential Spectrum, Dissertation, Belarusian State University at Minsk, 2010.
17. Yildirim, M., On the Spectrum of the Rhaly Operators on ℓ_{p}, Indian J. Pure Appl. Math., 32(2) (2001), 191-198.
18. Young, S. W., Algebraic and Spectral Properties of Generalized Cesàro Operators, Dissertation, University of North Carolina at Chapel Hill, 2002.
19. Young, S. W., Spectral Properties of Generalized Cesàro Operators, Integr. Equ. Oper. Theory, 50 (2004), 129-146.
20. Young, S. W., Generalized Cesàro Operators and the Bergman Spaces, J. Oper. Theory, 52 (2004), 341-351.