The Wikipedia page on the Hermitian adjoint is inconsistent about whether that operation is only defined for linear operators on a single Hilbert space, or more generally for arbitrary linear maps between (not necessarily identical) Hilbert spaces. It seems to me that the concept works equally well for general linear maps, but there may be some subtle technicality that requires restricting to linear operators.
Is there any reason why it's mathematically necessary to restrict to linear operators? If not, then is there any reason why doing so is more natural, convenient, or useful?