In this paper we study sequence spaces equations (SSE) with operators, which are determined by an identity whose each term 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 $s$.
References
A. Farés and B. de Malafosse, Sequence spaces equations and application to matrix transformationsInt. Math. Forum 3 (2008), no. 17-20, 911–927. MR2403407A. Farés and B. de Malafosse, Sequence spaces equations and application to matrix transformationsInt. Math. Forum 3 (2008), no. 17-20, 911–927. MR2403407
I.J. Maddox, Infinite Matrices of Operators, Springer-Verlag, Berlin, Heidelberg and New York, 1980. MR568707 0424.40002I.J. Maddox, Infinite Matrices of Operators, Springer-Verlag, Berlin, Heidelberg and New York, 1980. MR568707 0424.40002
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. MR1914961 1016.40002B. 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. MR1914961 1016.40002
B. de Malafosse, On some BK space, Int. J. Math. Math. Sci. 2003, no. 28, 1783–1801. MR1986671 1023.46010 10.1155/S0161171203204324B. de Malafosse, On some BK space, Int. J. Math. Math. Sci. 2003, no. 28, 1783–1801. MR1986671 1023.46010 10.1155/S0161171203204324
B. de Malafosse, Sum of sequence spaces and matrix transformations, Acta Math. Hung. 113 (3) (2006), 289–313. MR2286262 10.1007/s10474-006-0106-9B. de Malafosse, Sum of sequence spaces and matrix transformations, Acta Math. Hung. 113 (3) (2006), 289–313. MR2286262 10.1007/s10474-006-0106-9
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. MR2843150B. 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. MR2843150
B. de Malafosse and V. Rakočević, A generalization of a Hardy theorem, Linear Algebra Appl. 421 (2007), 306–314. MR2294343 1112.40005 10.1016/j.laa.2006.07.022B. de Malafosse and V. Rakočević, A generalization of a Hardy theorem, Linear Algebra Appl. 421 (2007), 306–314. MR2294343 1112.40005 10.1016/j.laa.2006.07.022
B. de Malafosse, Sum of sequence spaces and matrix transformations mapping in $s\sp 0\sb \alpha((\Delta-\lambda I)\sp h)+s\sp {(c)}\sb \beta((\Delta-\mu I)\sp l)$, Acta Math. Hung. 122 (2008), 217–230. MR2480862 1199.40035 10.1007/s10474-008-8008-7B. de Malafosse, Sum of sequence spaces and matrix transformations mapping in $s\sp 0\sb \alpha((\Delta-\lambda I)\sp h)+s\sp {(c)}\sb \beta((\Delta-\mu I)\sp l)$, Acta Math. Hung. 122 (2008), 217–230. MR2480862 1199.40035 10.1007/s10474-008-8008-7
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. MR1916930 1194.46006 10.1007/BF02871655B. 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. MR1916930 1194.46006 10.1007/BF02871655
B. de Malafosse and V. Rakočević, Applications of measure of noncompactness in operators on the spaces $s\sb \alpha,s\sp 0\sb \alpha,s\sp {(c)}\sb \alpha,l\sp p\sb \alpha$, J. Math. Anal. Appl. 323 (2006), 131–145. MR2261156 1106.47029 10.1016/j.jmaa.2005.10.024B. de Malafosse and V. Rakočević, Applications of measure of noncompactness in operators on the spaces $s\sb \alpha,s\sp 0\sb \alpha,s\sp {(c)}\sb \alpha,l\sp p\sb \alpha$, J. Math. Anal. Appl. 323 (2006), 131–145. MR2261156 1106.47029 10.1016/j.jmaa.2005.10.024
F. Móricz and B.E. Rhoades, An equivalent reformulation of summability by weighted mean methods, Linear Algebra Appl. 268 (1998), 171–181. MR1480339 10.1016/S0024-3795(97)00039-6F. Móricz and B.E. Rhoades, An equivalent reformulation of summability by weighted mean methods, Linear Algebra Appl. 268 (1998), 171–181. MR1480339 10.1016/S0024-3795(97)00039-6
F. Móricz and B.E. Rhoades, An equivalent reformulation of summability by weighted mean methods, revisited, Linear Algebra Appl. 349 (2002), 187–192. MR1903730 1052.40009 10.1016/S0024-3795(01)00604-8F. Móricz and B.E. Rhoades, An equivalent reformulation of summability by weighted mean methods, revisited, Linear Algebra Appl. 349 (2002), 187–192. MR1903730 1052.40009 10.1016/S0024-3795(01)00604-8
A. Wilansky, Summability Through Functional Analysis, North-Holland Mathematics Studies 85, 1984. MR738632 0531.40008A. Wilansky, Summability Through Functional Analysis, North-Holland Mathematics Studies 85, 1984. MR738632 0531.40008
