On Some Properties of K-g-Riesz Bases in Hilbert Spaces

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^{*}).

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