Numerically and spectrally optimal dual frames in Hilbert spaces

Finding a best dual frame that minimizes the reconstruction errors when erasures occur is a deep-rooted problem in frame theory. The primary purpose of this paper is to introduce a new measurement for constructing optimal duals. We consider the numerical radius of the error operator which has some advantages over the previous measures. Then we give several equivalent conditions for which the canonical dual and alternate duals are numerically optimal dual frames. Moreover, we obtain some relations between optimal duals, spectrally optimal duals and numerically optimal duals for erasures. Finally, we survey the effect of operator perturbation on spectrally optimal duals. This leads to finding some equivalent conditions under which the canonical dual is a spectrally optimal dual for any r-erasures.