Derived categories of Gushel-Mukai surfaces and Fano fourfolds of K3 type

We prove that very general, dual Gushel-Mukai surfaces are not isomorphic, though derived and L-equivalent. We use this result to study two semiorthogonal decompositions for a family of Fano fourfolds of K3 type, answering a question by Bernardara-Fatighenti-Manivel-Tanturri.