Document Type : Research Paper
Authors
- Chander .Shekhar ^{1}
- Sunayana Bhati ^{} ^{2}
- G.S. Rathore ^{2}
^{1} Department of Mathematics Indraprastha college for Women, University of Delhi, Delhi 110054, India.
^{2} Department of Mathematics and Statistics, University college of Science, M.L.S. University, Udaipur, Rajasthan, India.
Abstract
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 $ \mathcal{H} $ with respect to $ \mu $ 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.
Keywords
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