cK-frames and cK-Riesz bases in Hilbert spaces

In this paper, we prove some new results about ck-frames. also, we introduce the concept of ck-riesz basis and we provide a necessary and sufficient condition under which f is a ck-riesz basis. finally, for the closed range operator k ∈ b(h), we prove that under some conditions, πr(k)f is a ck-riesz basis if and only if it has only one dual, where πr(k) is the orthogonal projection from h onto r(k), i.e., the range of k.