{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T22:03:12Z","timestamp":1780524192978,"version":"3.54.1"},"reference-count":47,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,3]],"date-time":"2022-06-03T00:00:00Z","timestamp":1654214400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah","award":["305-130-1442"],"award-info":[{"award-number":["305-130-1442"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The sequence spaces \u2113p(\u2207q2)(0\u2264p&lt;\u221e) and \u2113\u221e(\u2207q2) are introduced by using the q-difference operator \u2207q2 of the second order. Apart from studying some basic properties of these spaces, we construct the basis and obtain the \u03b1-, \u03b2- and \u03b3-duals of these spaces. Besides some matrix classes involving q-difference sequence spaces, \u2113p(\u2207q2) and \u2113\u221e(\u2207q2) are characterized. The final section is devoted to classifying the spectrum of the q-difference operator \u2207q2 over the space \u21131 of absolutely summable sequences.<\/jats:p>","DOI":"10.3390\/sym14061155","type":"journal-article","created":{"date-parts":[[2022,6,3]],"date-time":"2022-06-03T08:01:18Z","timestamp":1654243278000},"page":"1155","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Sequence Spaces and Spectrum of q-Difference Operator of Second Order"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7780-8137","authenticated-orcid":false,"given":"Abdullah","family":"Alotaibi","sequence":"first","affiliation":[{"name":"Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3435-8417","authenticated-orcid":false,"given":"Taja","family":"Yaying","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Dera Natung Government College, Itanagar 791113, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9050-9104","authenticated-orcid":false,"given":"Syed Abdul","family":"Mohiuddine","sequence":"additional","affiliation":[{"name":"Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia"},{"name":"Department of General Required Courses, Mathematics, Faculty of Applied Studies, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ba\u015far, F. (2012). Summability Theory and Its Applications, Bentham Science Publishers.","DOI":"10.2174\/97816080545231120101"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"235","DOI":"10.5269\/bspm.v31i2.17541","article-title":"On the spectrum of 2nd order generalized difference operator \u03942 over the sequence space c0","volume":"31","author":"Dutta","year":"2013","journal-title":"Bol. Soc. Paran. Mat."},{"key":"ref_3","first-page":"18","article-title":"On some difference sequence spaces","volume":"17","author":"Et","year":"1993","journal-title":"Do\u01e7a-Tr. J. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1016\/j.na.2006.11.015","article-title":"On the fine spectrum of the generalized difference operator B(r,s) over the sequence spaces \u2113p and bvp, (1<p<\u221e)","volume":"68","author":"Furkan","year":"2008","journal-title":"Nonlinear Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1299","DOI":"10.1016\/j.camwa.2010.06.010","article-title":"Some new sequence spaces derived by the domain of generalized difference matrix","volume":"60","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"883","DOI":"10.1016\/j.mcm.2006.08.008","article-title":"On the fine spectrum of the operator B(r,s,t) over the sequence space \u21131 and bv","volume":"45","author":"Furkan","year":"2007","journal-title":"Math. Comput. Model."},{"key":"ref_7","first-page":"377","article-title":"On some generalized difference sequence spaces","volume":"21","author":"Et","year":"1995","journal-title":"Soochow J. Math."},{"key":"ref_8","first-page":"1776","article-title":"On the spectra of the generalized rth difference operator \u0394vr on the sequence space \u21131","volume":"219","author":"Dutta","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_9","first-page":"279","article-title":"On compact operators on the Riesz Bm-difference sequence space","volume":"A4","author":"Kara","year":"2011","journal-title":"Iran. J. Sci. Technol. Trans. A Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1016\/j.jmaa.2018.10.051","article-title":"The matrix domain and the spectra of a generalized difference operator","volume":"470","author":"Meng","year":"2019","journal-title":"J. Math. Anal. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1016\/j.jmaa.2003.12.006","article-title":"Generalized difference sequence spaces and their dual spaces","volume":"292","author":"Et","year":"2004","journal-title":"J. Math. Anal. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"339","DOI":"10.2478\/s12175-007-0028-1","article-title":"Generalized strongly difference convergent sequences associated with multiplier sequences","volume":"57","author":"Esi","year":"2007","journal-title":"Math. Slovaca"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"475","DOI":"10.2478\/s12175-007-0039-y","article-title":"On some new type generalized difference sequence spaces","volume":"57","author":"Esi","year":"2007","journal-title":"Math. Slovaca"},{"key":"ref_14","first-page":"25","article-title":"On K\u00f6the-Toeplitz duals of generalized difference seuence spaces","volume":"23","author":"Et","year":"2000","journal-title":"Bull. Malaysian Math. Sci. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1186\/s13662-020-03099-6","article-title":"Sequence spaces derived by the triple band generalized Fibonacci difference operator","volume":"2020","author":"Yaying","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1017\/S0080456800002751","article-title":"On q-functions and a certain difference operator","volume":"46","author":"Jackson","year":"1908","journal-title":"Trans. R. Soc. Edinb."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/s40995-019-00815-0","article-title":"Operators of basic (or q-) calculus and fractional q-calculus and their application in geometric function theory","volume":"44","author":"Srivastava","year":"2020","journal-title":"Iran. J. Sci. Technol. Trans. A Sci."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Kac, V., and Cheung, P. (2002). Quantum Calculus, Springer.","DOI":"10.1007\/978-1-4613-0071-7"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4717","DOI":"10.1016\/j.cam.2010.08.018","article-title":"q-Ces\u00e0ro matrix and q-statistical convergence","volume":"235","author":"Aktuglu","year":"2011","journal-title":"J. Comput. Appl. Math."},{"key":"ref_20","first-page":"8899960","article-title":"On generalized (p,q)-Euler matrix and associated sequence spaces","volume":"2021","author":"Yaying","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Yaying, T., Hazarika, B., Tripathy, B.C., and Mursaleen, M. (2022). The spectrum of second order quantum difference operator. Symmetry, 14.","DOI":"10.3390\/sym14030557"},{"key":"ref_22","unstructured":"Yaying, T., Kara, M.\u0130., Hazarika, B., and Kara, E.E. A study on q-analogue of Catalan sequence spaces. Filomat, accepted."},{"key":"ref_23","first-page":"97","article-title":"q-Ces\u00e0ro sequence spaces derived by q-analogues","volume":"5","author":"Demiriz","year":"2016","journal-title":"Adv. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"124453","DOI":"10.1016\/j.jmaa.2020.124453","article-title":"On sequence space derived by the domain of q-Ces\u00e0ro matrix in \u2113p space and the associated operator ideal","volume":"493","author":"Yaying","year":"2021","journal-title":"J. Math. Anal. Appl."},{"key":"ref_25","first-page":"35","article-title":"q-Hausdorff summability","volume":"7","author":"Bustoz","year":"2005","journal-title":"J. Comput. Anal. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"905","DOI":"10.2298\/FIL2203905Y","article-title":"Domain of Padovan q-difference matrix in sequence spaces \u2113p and \u2113\u221e","volume":"36","author":"Yaying","year":"2022","journal-title":"Filomat"},{"key":"ref_27","first-page":"387","article-title":"On the space of p-summable difference sequences of order m, (1\u2264p<\u221e)","volume":"43","author":"Altay","year":"2006","journal-title":"Studia Sci. Math. Hungar."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1524\/anly.1997.17.1.87","article-title":"Matrix transformations in spaces of bounded and convergent difference sequences of order m","volume":"17","author":"Malkowsky","year":"2007","journal-title":"Analysis"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"169","DOI":"10.4153\/CMB-1981-027-5","article-title":"On certain sequence spaces","volume":"24","year":"1981","journal-title":"Canad. Math. Bull."},{"key":"ref_30","first-page":"11","article-title":"A new type of difference sequence spaces","volume":"1","author":"Tripathy","year":"2016","journal-title":"Int. J. Sci. Technol."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01215107","article-title":"Matrixtransformationen von Folgenr\u00e4umen eine Ergebnis\u00fcbersicht","volume":"154","author":"Stieglitz","year":"1977","journal-title":"Math. Z."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1006\/jmaa.1993.1398","article-title":"Matrix transformations between the sequence spaces of Maddox","volume":"180","year":"1993","journal-title":"J. Math. Anal. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1017\/S0305004100001109","article-title":"Matrix transformations between some classes of sequences","volume":"68","author":"Lascarides","year":"1970","journal-title":"Proc. Camb. Philos. Soc."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1757","DOI":"10.1007\/s10114-005-0777-0","article-title":"On the fine spectra of the difference operator \u0394 over the sequence space bvp, (1\u2264p<\u221e)","volume":"23","author":"Akhmedov","year":"2007","journal-title":"Acta Math. Sin. Engl. Ser."},{"key":"ref_35","first-page":"586","article-title":"The fine spectra of the difference operator \u0394 over the sequence space \u2113p, (1\u2264p<\u221e)","volume":"39","author":"Akhmedov","year":"2006","journal-title":"Demonstr. Math."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/j.ins.2004.02.007","article-title":"On the fine spectrum of the difference operator \u0394 on c0 and c","volume":"168","author":"Altay","year":"2004","journal-title":"Inf. Sci."},{"key":"ref_37","first-page":"1","article-title":"The fine spectrum and the matrix domain of the difference operator \u0394 on the sequence space \u2113p,0<p<1","volume":"2","author":"Altay","year":"2007","journal-title":"Comm. Math. Anal."},{"key":"ref_38","first-page":"1153","article-title":"The fine spectra of the difference operator \u0394 over the sequence spaces \u21131 and bv","volume":"1","author":"Kayaduman","year":"2006","journal-title":"Int. Math. For."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"893","DOI":"10.14492\/hokmj\/1285766434","article-title":"On the fine spectrum of the generalized difference operator B(r,s) over the sequence space \u21131 and bv","volume":"35","author":"Furkan","year":"2006","journal-title":"Hokkaido Math. J."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"989","DOI":"10.1016\/j.camwa.2006.07.006","article-title":"On the fine spectrum of the operator B(r,s,t) over c0 and c","volume":"53","author":"Furkan","year":"2007","journal-title":"Comput. Math. Appl."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2141","DOI":"10.1016\/j.camwa.2010.07.059","article-title":"On the fine spectrum of the operator B(r,s,t) over the sequence spaces \u2113p and bvp, (1<p<\u221e)","volume":"60","author":"Furkan","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1007\/s13398-020-00997-y","article-title":"A survey on the spectra of the difference operators over the Banach space c","volume":"115","author":"Baliarsingh","year":"2021","journal-title":"RACSAM"},{"key":"ref_43","first-page":"555","article-title":"On a spectral classification of the operator \u0394vr over the sequence space c0","volume":"84","author":"Baliarsingh","year":"2014","journal-title":"Proc. Nat. Acad. Sci. India"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1007\/s40995-020-00871-x","article-title":"On a generalized difference operator and its fine spectra","volume":"44","author":"Baliarsingh","year":"2020","journal-title":"Iran J. Sci. Technol. Trans. Sci."},{"key":"ref_45","unstructured":"Wilansky, A. (1984). Summability through Functional Analysis, Elsevier. North-Holland Mathematics Studies."},{"key":"ref_46","unstructured":"Goldberg, S. (1985). Unbounded Linear Operators, Dover Publications, Inc."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"7041","DOI":"10.1007\/s00521-018-3521-2","article-title":"An approach towards decision-making and medical diagnosis problems using the concept of spherical fuzzy sets","volume":"31","author":"Mahmood","year":"2019","journal-title":"Neural Comput. Applic."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1155\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:24:08Z","timestamp":1760138648000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1155"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,3]]},"references-count":47,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["sym14061155"],"URL":"https:\/\/doi.org\/10.3390\/sym14061155","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,3]]}}}