On the spectrum and Hilbert Schimidt properties of generalized Rhaly matrices

Year 2019, Volume: 68 Issue: 1, 712 - 723, 01.02.2019

Mohammad Mursaleen , Mustafa Yildirim , Nuh Durna

https://doi.org/10.31801/cfsuasmas.464177

Cited By: 8

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

• 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.
• Aleman, A. and Siskakis, A. G., An integral Operator on H^{p}, Complex Variables, 28 (1995), 149-158.
• Brown, A., Halmos, P. R. and Shields, A. L., Cesàro Operators, Acta Sci. Math., 26 (1965), 125-134.
• Cima, J. A. and Petersen, K .E., Some Analytic Functions Whose Boundary Values Have Bounded Mean Oscillation, Math. Z., 147 (1976), 237-247.
• Conway, J. B., A Course in Operator Theory, AMS, 2000.
• Durna, N. and Yildirim, M., Topological resuls about the set of generalized Rhally matrices, Gen. Math. Notes, 17(1) (2013), 1-7.
• Durna, N. and Yildirim, M., Generalized terraced matrices, Miskolc Math. Notes, 17(1) (2016), 201-208.
• Garnett, J. B., Bounded analytic functions, vol. 96 of Pure and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981.
• Gonzalez, M., The fine spectrum of the Cesàro operator in ℓ_{p} (1<p<∞), Arch. Math., 44 (1985), 355-358.
• Hardy, G. H., Littlewood, J. E. and Polya, G., Inequalities, Cambridge, 1934.
• Okutoyi, I. J., On the spectrum of C₁ as an operator on bv₀, J. Austral. Math. Soc. Series A, 48 (1990), 79-86.
• Pommerenke, C., Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation, Comment. Math. Helv., 52 (1977), 591-602.
• Rhaly, Jr. H. C., Terraced matrices, Bull. London Math. Soc., 21 (1989), 399-406.
• Rhoades, B. E. and Yildirim, M., The spectra of factorable matrices on ℓ_{p}, Integr. Equ. Oper. Theory, 55 (2006), 111-126.
• Siskakis, A. G., Composition Semigroups and the Cesàro Operator on H^{p}, J. London Math. Soc., 36 (1987), 153-164.
• Valeriyevna, G. O., Strength Properties of Safara Essential Spectrum, Dissertation, Belarusian State University at Minsk, 2010.
• Yildirim, M., On the Spectrum of the Rhaly Operators on ℓ_{p}, Indian J. Pure Appl. Math., 32(2) (2001), 191-198.
• Young, S. W., Algebraic and Spectral Properties of Generalized Cesàro Operators, Dissertation, University of North Carolina at Chapel Hill, 2002.
• Young, S. W., Spectral Properties of Generalized Cesàro Operators, Integr. Equ. Oper. Theory, 50 (2004), 129-146.
• Young, S. W., Generalized Cesàro Operators and the Bergman Spaces, J. Oper. Theory, 52 (2004), 341-351.