Matrix transformations and sequence spaces equations. (English) Zbl 1270.40008

Summary: We study sequence spaces equations (SSE) with operators, which are determined by an identity, each term of which is a sum or a sum of products of sets of the form \(\chi _{a}\left( T\right)\) and \(\chi _{f\left( x\right) }\left( T\right)\) where \(f\) maps \(U^{+}\) to itself, \(\chi \) is either of the symbols \(s, s^{0}\), or \(s^{\left( c\right) }\). Then we solve five (SSE) of the form \(\chi _{a}+\chi _{x}^{\prime }=\chi _{b}^{\prime }\), where \(\chi , \chi ^{\prime }\) are either \(s^{{{}0}}, s^{\left( c\right) }\), or \(s\). We apply the previous results to the solvability of the systems \(s_{a}^{0}+s_{x}\left( \Delta \right) =s_{b}\), \(s_{x}\supset s_{b}\), and \(s_{a}+s_{x}^{\left( c\right) }\left( \Delta \right) =s_{b}^{\left( c\right) }\), \(s_{x}^{\left( c\right) }\supset s_{b}^{\left( c\right) }\). Finally, we solve the (SSE) with operators defined by \(\chi _{a}\left( C\left( \lambda \right) D_{\tau }\right) +s_{x}^{\left( c\right) }\left( C\left( \mu \right) D_{\tau }\right) =s_{b}^{\left( c\right) }\) where \(\chi \) is either \(s^{0}\), or \(s\).


40H05 Functional analytic methods in summability
46A45 Sequence spaces (including Köthe sequence spaces)
Full Text: DOI Euclid


[1] A. Farés and B. de Malafosse, Sequence spaces equations and application to matrix transformations Int. Math. Forum 3 (2008), no. 17-20, 911-927. · Zbl 1168.40003
[2] G.H. Hardy, Divergent Series , Oxford University Press, Oxford, 1949. · Zbl 0032.05801
[3] I.J. Maddox, Infinite Matrices of Operators, Springer-Verlag, Berlin, Heidelberg and New York, 1980. · Zbl 0424.40002
[4] B. de Malafosse, Properties of some sets of sequences and application to the spaces of bounded difference sequences of order \(\mu \), Hokkaido Math. J. 31 (2002), 283-299. · Zbl 1016.40002
[5] B. de Malafosse, On some BK space , Int. J. Math. Math. Sci. 2003 , no. 28, 1783-1801. · Zbl 1023.46010
[6] B. de Malafosse, Sum of sequence spaces and matrix transformations, Acta Math. Hung. 113 ( 3 ) (2006), 289-313. · Zbl 1121.40010
[7] B. de Malafosse, Application of the infinite matrix theory to the solvability of certain sequence spaces equations with operators. Mat. Vesnik 54 (2012), no. 1, 39-52. · Zbl 1349.40017
[8] B. de Malafosse, Solvability of certain sequence spaces inclusion equations with operators, Demonstratio Math. (to appear). · Zbl 1307.40009
[9] B. de Malafosse and V. Rakočević, A generalization of a Hardy theorem , Linear Algebra Appl. 421 (2007), 306-314. · Zbl 1112.40005
[10] B. de Malafosse, Sum of sequence spaces and matrix transformations mapping in \(s^ 0_ \alpha((\Delta-\lambda I)^ h)+s^ {(c)}_ \beta((\Delta-\mu I)^ l)\) , Acta Math. Hung. 122 (2008), 217-230. · Zbl 1199.40035
[11] B. de Malafosse and E. Malkowsky, Sequence spaces and inverse of an infinite matrix , Rend. del Circ. Mat. di Palermo Serie II 51 (2002), 277-294. · Zbl 1194.46006
[12] B. de Malafosse and E. Malkowsky, Sets of difference sequences of order \(m\) , Acta Sci. Math. (Szeged) 70 (2004), 659-682. · Zbl 1075.46004
[13] B. de Malafosse and V. Rakočević, Applications of measure of noncompactness in operators on the spaces \(s_ \alpha,s^ 0_ \alpha,s^ {(c)}_ \alpha,l^ p_ \alpha\) , J. Math. Anal. Appl. 323 (2006), 131-145. · Zbl 1106.47029
[14] F. Móricz and B.E. Rhoades, An equivalent reformulation of summability by weighted mean methods, Linear Algebra Appl. 268 (1998), 171-181. · Zbl 0897.40003
[15] F. Móricz and B.E. Rhoades, An equivalent reformulation of summability by weighted mean methods, revisited , Linear Algebra Appl. 349 (2002), 187-192. · Zbl 1052.40009
[16] A. Wilansky, Summability Through Functional Analysis , North-Holland Mathematics Studies 85, 1984. · Zbl 0531.40008
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.