Q-duals and Q-approximate duals of g-frames in Hilbert spaces

In this paper we mainly discuss the properties of Q-duals and Q-approximate duals of g-frames in Hilbert spaces. Given {Λj}j∈J and {Θj}j∈J being a pair of Q-dual, {Γj}j∈J being some kind of perturbed sequence of {Θj}j∈J, in general {Λj}j∈J is not a Q-approximate dual of {Γj}j∈J. We then give four different kinds of perturbed conditions such that {Λj}j∈J and {Γj}j∈J, a perturbed sequence of {Θj}j∈J, are possible to be a pair of Q-approximate dual. We also provide several different methods to construct Q-duals and Q-approximate duals of g-frames. Finally, we give two equivalent characterizations of Q-duals and Q-approximate duals by using the associated induced sequences.