×

The roots of polynomials and the operator \(\Delta_i^3\) on the Hahn sequence space \(h\). (English) Zbl 1476.46003

Summary: In this paper, we define the third order generalized difference operator \(\Delta_i^3\), where \[ (\Delta_i^3x)_k=\sum_{i=0}^3\frac{(-1)^i}{i+1}\binom{3}{i} x_{k-i}= x_k-\frac{3}{2}x_{k-1}+x_{k-2}-\frac{1}{4}x_{k-3}, \] and show that it is a linear bounded operator on the Hahn sequence space \(h\). Then we study the spectrum and point spectrum of the operator \(\Delta_i^3\) on \(h\). Furthermore, we determine the point spectrum of the adjoint of this operator. This is achieved by studying some properties of the roots of certain third order polynomials.

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
40C05 Matrix methods for summability
46B45 Banach sequence spaces
47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aasma, A.; Dutta, H.; Natarajan, PN, An introductory course in summability theory (2017), Hoboken: Wiley, Hoboken · Zbl 1375.40001 · doi:10.1002/9781119397786
[2] Başar, F.; Dutta, H., Summable spaces and their duals, matrix transformations and geometric properties (2020), Boca Raton: Chapman and Hall/CRC, Boca Raton · Zbl 1478.46001 · doi:10.1201/9781351166928
[3] Baliarsingh, P.; Dutta, S., On certain Toeplitz matrices via difference operator and their applications, Afr Math, 27, 781-793 (2016) · Zbl 1369.40006 · doi:10.1007/s13370-015-0374-z
[4] Choudhary, B.; Nanda, S., Functional analysis with applications (1989), New York: Wiley, New York · Zbl 0698.46001
[5] Das, R., On the fine spectrum of the lower triangular matrix \({B(r, s)}\) over the Hahn sequence space, Kyungpook Math J, 57, 3, 441-455 (2017) · Zbl 1489.47006
[6] Dutta, H.; Peters, JF, Applied mathematical analysis: theory, methods and applications, volume 177 of studies in system, decision and control, chapter on the spectra of difference operators over some Banach spaces, 791-810 (2020), New York: Springer, New York · Zbl 1423.47001
[7] Dutta, H.; Rhoades, BE, Current topics in summability theory and applications (2016), Singapore: Springer, Singapore · Zbl 1348.40001
[8] Hahn, H., Über folgen linearer Operationen, Monatshefte für Mathematik und Physik, 32, 1, 3-88 (1922) · JFM 48.0473.01 · doi:10.1007/BF01696876
[9] Kirişci, M., The Hahn sequence space defined by the Cesàro mean, Abstract and applied analysis (2013), London: Hindawi, London · Zbl 1334.46007
[10] Kirişci, M., A survey on Hahn sequence space, Gen Math Notes, 19, 2, 37-58 (2013)
[11] Kirişci M \((2014) p\)-Hahn sequence space. arXiv preprint arXiv:1401.2475 · Zbl 1328.46005
[12] Malkowsky, E.; Rakočević, V., Advanced functional analysis (2019), Boca Raton: CRC Press, Boca Raton · Zbl 1468.46001 · doi:10.1201/9780429442599
[13] Malkowsky E, Rakočević V, Tuǧ O (2021) Compact operators on the Hahn space. Monatsh Math. doi:10.1007/s00605-021-01588-8 · Zbl 1480.46003
[14] Mursaleen, M.; Başar, F., Sequence spaces, topics in modern summability (2020), Boca Raton: CRC Press, Boca Raton · Zbl 1468.46003 · doi:10.1201/9781003015116
[15] Rao, KC, The Hahn sequence spaces I, Bull Calcutta Math Soc, 82, 72-78 (1990) · Zbl 0719.46003
[16] Rao, KC; Srinivasalu, TG, The Hahn sequence spaces II, Y.Y.U. J Faculty Educ, 2, 43-45 (1996)
[17] Rao KC, Subramanian N (2002) The Hahn sequence space III. Bull Malays Math Sci Soc 25(2):163-171 · Zbl 1186.46020
[18] Sawano, Y.; El-Shabrawy, SR, Fine spectra of the discrete generalized Cesàro operator on Banach sequence spaces, Monatsh Math, 192, 185-224 (2020) · Zbl 1443.47005 · doi:10.1007/s00605-020-01376-w
[19] Taylor, AE; Lay, DC, Introduction to functional analysis (1986), Malabar: Robert E. Krieger Publishing Company, Malabar · Zbl 0654.46002
[20] Yeşilkayagil, M.; Kirişci, M., On the fine spectrum of the forward difference operator on the Hahn space, Gen Math Notes, 33, 2, 1-16 (2016)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.