### Operator Lipschitz estimate functions on Banach spaces

#### Abstract

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).

**Published:**2019-09-17

**How to Cite this Article:**Simon Joseph, Arafa Dawood, Nagat Suoliman, Fatin Saeed, Mohammed Mustafa, Operator Lipschitz estimate functions on Banach spaces, Journal of Semigroup Theory and Applications, Vol 2019 (2019), Article ID 7 Copyright © 2019 Simon Joseph, Arafa Dawood, Nagat Suoliman, Fatin Saeed, Mohammed Mustafa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Journal of Semigroup Theory and Applications

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