On matrix domains of triangles. (English) Zbl 1132.46011

The authors’ summary clearly explains the content of this paper: “We prove some general results for the determination of the \(\beta \)-duals of, and the characterisations of matrix transformations on matrix domains of arbitrary triangles in FK-spaces. Our results contain almost all recently published ones as special cases. We also study the measure of noncompactness of several matrix transformations. In particular, we obtain some known results of J.J.Sember [J. Lond.Math.Soc., II.Ser.2, 530–534 (1970; Zbl 0199.11302)], give the corrected version of a recent result by I.Djolović [J. Math.Anal.Appl.318, No.2, 658–666 (2006; Zbl 1099.47021)], and study compact operators on some spaces of B.Altay, F.Başar and M.Mursaleen [Ukr.Mat.Zh.57, No.1, 3–17 (2005; Zbl 1096.46011); Inf.Sci.176, No.10, 1450–1462 (2006; Zbl 1101.46015); Nonlinear Anal., Theory Methods Appl.65, No.3(A), 707–717 (2006; Zbl 1108.46019)].”


46B45 Banach sequence spaces
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B07 Local theory of Banach spaces
Full Text: DOI


[1] Akhmerov, R. R.; Kamenskii˘, M. I.; Potapov, A. S.; Rodkina, A. E.; Sadovskii˘, B. N., Measures of Noncompactness and Condensing Operators Operator Theory: Advances and Applications, vol. 55 (1992), Birkhäuser: Birkhäuser Basel · Zbl 0748.47045
[2] Altay, B.; Başar, F., On the paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math., 26, 5, 701-715 (2002) · Zbl 1058.46002
[3] Altay, B.; Başar, F., Some Euler sequence spaces of non-absolute type, Ukr. Math. J., 57, 1, 1-17 (2005) · Zbl 1096.46011
[4] Altay, B.; Başar, F.; Mursaleen, M., On the Euler sequence spaces which include the spaces \(ℓ_p\) and \(ℓ_∞ I\), Inform. Sci., 176, 1450-1462 (2006) · Zbl 1101.46015
[5] Altay, B.; Başar, F.; Mursaleen, M., On the Euler sequence spaces which include the spaces \(ℓ_p\) and \(ℓ_∞\) II, Nonlinear Anal., 65, 707-717 (2006) · Zbl 1108.46019
[6] Aydın, C.; Başar, F., Some new difference sequence spaces, Appl. Math. Comput., 157, 677-693 (2004) · Zbl 1072.46007
[7] Banaś, J.; Goebel, K., Measures of Noncompactness in Banach Spaces. Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, vol. 60 (1980), Marcel Dekker: Marcel Dekker New York · Zbl 0441.47056
[8] Başar, F.; Šengönul, M., Some new Ceàro sequence spaces of non-absolute type, Soochow J. Math., 31, 1, 107-119 (2005) · Zbl 1085.46500
[9] Cooke, R. C., Infinite Matrices and Sequence Spaces (1950), MacMillan and Co. Ltd: MacMillan and Co. Ltd London · Zbl 0040.02501
[10] Djolović, I., Compact operators on the spaces \(a_0^r(\Delta)\) and \(a_c^r(\Delta)\), J. Math. Anal. Appl., 318, 658-666 (2006) · Zbl 1099.47021
[11] Gohberg, I. T.; Goldenstein, L. S.; Markus, A. S., Investigations of some properties of bounded linear operators with their \(q\)-norms, Učen. Zap. Kishinevsk. Univ., 29, 29-36 (1957), (Russian)
[12] Jarrah, A. M.; Malkowsky, E., Ordinary absolute and strong summability and matrix transformations, Filomat, 17, 59-78 (2003) · Zbl 1274.40001
[13] Kızmaz, H., On certain sequence spaces, Can. Math. Bull., 24, 169-176 (1981) · Zbl 0454.46010
[14] de Malafosse, B.; Rakočević, V., Applications of measure of noncompactness in operators on the spaces \(s_\alpha, s_\alpha^o, s_\alpha^{(c)}, \ell_\infty^p\), J. Math. Anal. Appl., 323, 131-145 (2006) · Zbl 1106.47029
[15] de Malafosse, B.; Malkowsky, E.; Rakočević, V., Measure of noncompactness of operators and matrices on the spaces \(c\) and \(c_0\), Int. J. Math. Math. Sci. (2006), Art. ID 46930, 5 pp · Zbl 1154.47027
[16] Malkowsky, E.; Rakočević, V., An introduction into the theory of sequence spaces and measures of noncompactness, Zbornik radova, Matematički institut SANU, Beograd, 9, 17, 143-243 (2000)
[17] Malkowsky, E.; Rakočević, V.; Živković, S., Matrix transformations bewteen the sequence space \(bv^p\) and certain BK spaces, Bulletin T. CXXXIII de l’Académie Serbe des Sciences et Arts, Classe des Sciences mathématiques et naturelles, Sciences Mathématiques, 27, 33-46 (2002) · Zbl 1033.46002
[18] Ng, P.-N.; Lee, P.-Y., Cesàro sequence spaces of non-absolute type, Comment. Math. Prace. Mat., 20, 2, 429-433 (1978) · Zbl 0408.46012
[19] Sember, J. J., Summability matrices as compact-like operators, J. London Math. Soc., 2, 2, 534-539 (1970) · Zbl 0199.11302
[20] Sirajudeen, S. M., Matrix transformations of bv into \(\ell(q), \ell_\infty(q), c_0(q)\) and \(c(q)\), Indian J. Pure Appl. Math., 23, 1, 55-61 (1992) · Zbl 0767.40003
[21] Stieglitz, M.; Tietz, H., Matrixtransformationen von Folgenräumen. Eine Ergebnisübersicht, Math. Z., 154, 1-16 (1977) · Zbl 0331.40005
[22] Wang, C.-S., On Nörlund sequence spaces, Tamkang J. Math., 9, 269-279 (1978)
[23] Wilansky, A., Functional Analysis (1964), Blaisdel Publishing Co.: Blaisdel Publishing Co. New York, Toronto, London · Zbl 0136.10603
[24] Wilansky, A., Summability Through Functional Analysis. Summability Through Functional Analysis, Mathematics Studies, 85 (1984), North-Holland: North-Holland Amsterdam · Zbl 0531.40008
[25] Zeller, K., Faktorfolgen bei Limitierungsverfahren, Math. Z., 56, 134-151 (1952) · Zbl 0046.06402
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.