Wavelet and Linear Algebra

Abstract 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 $K \in B(\mathcal{H})$, we prove that under some conditions, $\pi_{R(K)}F$ is a cK-Riesz basis if and only if it has only one dual, where $\pi_{R(K)}$ is the orthogonal projection from $\mathcal{H}$ onto $R(K)$, i.e., the range of $K$.

Abstract In this paper, we study the K-Riesz bases and the K-g-Riesz bases in Hilbert spaces. We show that for $K \in B(\mathcal{H})$, a K-Riesz basis is precisely the image of an orthonormal basis under a bounded left-invertible operator such that the range of this operator includes the range of $K$. Also, we show that $\lbrace \Lambda_i \in B(\mathcal{H}, \mathcal{H}_i ) : \, i \in I \rbrace$ is a K-g-Riesz basis for $\mathcal{H}$ with respect to $\lbrace \mathcal{H}_i \rbrace_{i \in I}$
if and only if there exists a g-orthonormal basis $\lbrace Q_i \rbrace_{i \in I}$
for $\mathcal{H}$ and a bounded right-invertible operator $U $ on $\mathcal{H}$
such that $\Lambda_i = Q_i U$ for all $i \in I$, and $R(K) \subset R(U^{*})$.