COLLOQUIUM MATHEMATICUM Online First version INTERPOLATIVE LIPSCHITZ IDEALS BY MANAF ADNAN SALEH SALEH (Baghdad) Abstract. Building upon the interpolation procedure of H. Jarchow and U. Matter for linear operator ideals we define interpolative Lipschitz ideals between metric spaces and Banach spaces. We establish that the resulting class of Lipschitz operators is an injective Banach Lipschitz ideal and show several standard basic properties of that class. We extend the notion of (p, θ, q, ν )-dominated operator ideal to the Lipschitz setting, we prove the domination theorem in this case and establish several characterizations. Finally, we generalize the interpolative Lipschitz ideal procedure to arbitrary metric spaces and show the Lipschitz ideal properties for this notion. 1. Notations and preliminaries. The letters E, F and G will denote Banach spaces, and X and Y will stand for pointed metric spaces. The closed unit ball of a Banach space E is denoted by B E , and E ∗ is the dual space of E. The class of all bounded linear operators between arbitrary Banach spaces will be denoted by L. The symbols R + , R, and N stand for the sets of all positive real numbers, of all real numbers and of all positive integers, respectively. A map T from X into Y is called Lipschitz if there is a nonnegative constant C such that d Y (Tx 1 ,Tx 2 ) ≤ Cd X (x 1 ,x 2 ) for all x 1 , x 2 in X . The smallest possible C is the Lipschitz constant of T , denoted by Lip(T ). The space of all Lipschitz maps between X and Y is denoted by Lip(X, Y ). If Y = F , then Lip(X, F ) is a Banach space under the Lipschitz norm Lip(·). The members of Lip(X, F ) are known as Lipschitz operators. The Banach space of real-valued Lipschitz functions defined on X that send the special point 0 to 0, equipped with the Lipschitz norm Lip(·), will be denoted by X # . The space X # is called the Lipschitz dual of X . The symbols W (B E ∗ ) and W (B X #) stand for the set of all Borel probability measures defined on B E ∗ and B X #, respectively. The value of a functional a at an element x is denoted by 〈x, a〉. 2020 Mathematics Subject Classification : Primary 47L20; Secondary 26A16, 47A57. Key words and phrases : operator ideals, Lipschitz classes, operator methods in interpola- tion. Received 17 February 2019; revised 21 September 2019 and 20 May 2020. Published online 6 July 2020. DOI: 10.4064/cm7844-10-2019 [1] c  Instytut Matematyczny PAN, 2020