In this talk, we define and study a class of operators between Banach function spaces that extends the classical notion of diagonal (multiplication) operators. These maps, called lattice Lipschitz operators, $T: X \to Y$, satisfy pointwise Lipschitz estimates of the form
\[ |Tf(\omega)-Tg(\omega)| \leq K(\omega),|f(\omega)-g(\omega)|, \quad \forall \omega \in \Omega, \]
for a suitable measurable bound function.

This framework provides a natural nonlinear analogue of multiplication operators, preserving the underlying lattice structure and showing that, under some conditions, they can be represented as superposition operators. We present results from the finite-dimensional case to the setting where $X, Y$ are $C(K)$ spaces or Banach function spaces. Some examples, as well as factorization, representation, and extension results, are also discussed.

This talk is based on joint work with J.M. Calabuig, E. Erdoğan and E.A. Sánchez Pérez.