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 $R(K) subset R(U^{*})$.