  In this paper, we investigate direct and inverse problems for the
time-fractional heat equation with a time-dependent leading coefficient for
positive operators. First, we consider the direct problem, and the unique
existence of the generalized solution is established. We also deduce some
regularity results. Here, our proofs are based on the eigenfunction expansion
method. Second, we study the inverse problem of determining the leading
coefficient, and the well-posedness of this inverse problem is proved.
