  In this paper, we discuss the uniqueness for solution to time-fractional
diffusion equation $\partial_t^\alpha (u-u_0) + Au=0$ with the homogeneous
Dirichlet boundary condition, where an elliptic operator $-A$ is not
necessarily symmetric. We prove that the solution is identically zero if its
normal derivative with respect to the operator $A$ vanishes on an arbitrary
small part of the spatial domain over a time interval. The proof is based on
the Laplace transform and the spectral decomposition, and is valid for more
general time-fractional partial differential equations, including those
involving non symmetric operators.
