We consider the concept of a $ t $ -basis generated by the tensor product of Banach spaces.We introduce the concept of a $ * $ -harmonic Banach-valued ( $ X $ -valued) function, where the space $ X $ under consideration has an involution $ * $ .Using the fact that the classical system of exponentials forms a $ t $ -basis for the Bochner space of $ X $ -valued functions with the UMD property, we define the Hardy class of $ * $ -harmonic functions in the unit disk.We consider a boundary value problem with oblique derivative in this class and establish a criterion for the Fredholmness and, in particular, the solvability of this problem.Concrete examples are given.
