Generalized Continuous Frames for Operators

In this note, the notion of generalized continuous K-frame in a Hilbert space is defined. Examples have been given to exhibit the existence of generalized continuous K-frames. A necessary and sufficient condition for the existence of a generalized continuous K-frame in terms of its frame operator is obtained and a characterization of a generalized continuous K-frame for H with respect to μ is given. Also, a sufficient condition for a generalized continuous K-frame is given. Further, among other results, we prove that generalized continuous K-frames are invariant under a linear homeomorphism. Finally, keeping in mind the importance of perturbation theory in various branches of applied mathematics, we study perturbation of K-frames and obtain conditions for the stability of generalized continuous K-frames.