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Let (A(t))t∈R be a continuous family of unbounded operators on a UMD Banach space X, with t-independent domain W dense and compactly embedded in X. Under these and other general technical conditions, we prove that the operator DA = d dt − A(·) is a Fredholm operator between the spaces W (R, X) ∩ L(R, W ) and L(R, X) for every p ∈ (1,∞). We also characterize the index of DA by the spectral flow of A. These results generalize those obtained by Robbin and Salamon when X and W are Hilbert spaces, p = 2 and the family (A(t))t∈R is selfadjoint. The hypotheses involved are satisfied by broad classes of second order elliptic differential operators with boundary conditions and by PDE systems with a Hamiltonian-like structure. An application to the L maximal regularity for the nonautonomous Cauchy problem on the half-line is discussed in detail.

The Robbin-Salamon index theorem in Banach spaces with UMD