Some norm estimates for the triangular projection on the space of bounded linear operators between two symmetric sequence spaces

For two given symmetric sequence spaces E and F we study the action of the main triangular projection T on B(E, F), the space of all bounded linear operators from E to F, and give a lower estimate for the norm of T in terms of the fundamental functions of E and F and the fundamental function of the generalized dual space F : E. In addition, we give a condition for the boundedness of the operator T in terms of F-absolutely summing operators. Furthermore, we apply our results to some concrete symmetric sequence spaces. In particular, we study the question of the boundedness or unboundedness of T on the spaces \(B(\ell _{p_1,q_1},\ell _{p}),\) $B({\ell }_{p_1,q_1},{\ell }_p),$ \(B(\ell _{p},\ell _{p_1,q_1})\) $B({\ell }_p,{\ell }_{p_1,q_1})$ and \(B(\ell _{p_1,q_1},\ell _{p_2,q_2})\) $B({\ell }_{p_1,q_1},{\ell }_{p_2,q_2})$ .