We fully characterize the well-posedness in vector-valued Hölder in time spaces of a nonlocal equation involving a closed operator matrix \(\mathcal {A}\) with diagonal domain, defined on a product of Banach spaces, solely in terms of the norm boundedness of a block-operator-valued symbol. We also give vector-valued a priori maximal Hölder inequalities. Our result also contains a characterization in the case of a single closed linear operator (not necessarily bounded), which is also new. We show that the condition that \(\mathcal {A}\) is the generator of an analytic semigroup is sufficient for the well-posedness. In particular, we show that the well-posedness holds if the operators on the diagonal of \(\mathcal {A}\) are generators of analytic semigroups and if the off-diagonal entries satisfy a smallness condition. We exemplify our main results with abstract as well as concrete models arising in fluid dynamics.
