Nonlinear Operator Ideals Between Metric Spaces and Banach Spaces (PART I)

In this paper we present part I of nonlinear operator ideals theory between metric spaces and Banach spaces. Building upon the definition of operator ideal between arbitrary Banach spaces of A. Pietsch we pose three types of nonlinear versions of operator ideals. We introduce several examples of nonlinear ideals and the relationships between them. For every space ideal $\mathsf{A}$ can be generated by a special nonlinear ideal which consists of those Lipschitz operators admitting a factorization through a Banach space $\mathbf{M}\in\mathsf{A}$. We investigate products and quotients of nonlinear ideals. We devote to constructions three types of new nonlinear ideals from given ones. A "new" is a rule defining nonlinear ideals $\mathfrak{A}^{L}_{new}$, $\textswab{A}^{L}_{new}$, and $\textfrak{A}^{L}_{new}$ for every $\mathfrak{A}$, $\textswab{A}^{L}$, and $\textfrak{A}^{L}$ respectively, are called a Lipschitz procedure. Considering the class of all stable objects for a given Lipschitz procedure we obtain nonlinear ideals having special properties. We present the concept of a (strongly) $p-$Banach nonlinear ideal ($0<p<1$) and prove that the nonlinear ideals of Lipschitz nuclear operators, Lipschitz Hilbert operators, products and quotient are strongly $r-$Banach nonlinear ideals ($0<r<1$).