Operator Lipschitz estimate functions on Banach spaces

In this paper, let X, Y be Banach spaces and let ℒ(X, Y) be the space of bounded linear sequence of operators from X to Y. We develop the theory of double sequence of operators integrals on ℒ(X, Y) and apply this theory to obtain commutator series estimates, for a large class of functions 𝑓𝑗 , where 𝐴𝑗 ∈ ℒ(𝑋), B𝑗 ∈ ℒ(𝑌) are scalar type the sequence of operators and 𝑆 ∈ ℒ(𝑋, 𝑌). In particular, we establish this estimate for 𝑓𝑗 (1 + 𝜖): = |1 + 𝜖| and for diagonalizable estimates derive hold for diagonalizable matrices with a constant independent of the size of the sequence of operators on 𝑋 = l(1+𝜖) and 𝑌 = l(1+𝜖) , for 𝜖 = 0, and X = Y = c0. Also, we obtain results for 𝜖 ≥ 0, studied the estimate above [1] in the setting of Banach ideals in ℒ(𝑋, 𝑌).