Controlled g-bases for operators and some characterizations of them

In this note, we first define the concepts of controlled $K$-g-orthonormal bases and controlled $K$-g-Riesz bases in Hilbert spaces. In the sequel, we show that a sequence is a controlled $K$-g-orthonormal basis (resp. controlled $K$-g-Riesz basis) if and only if its induced sequence is a controlled $K$-orthonormal basis (resp. controlled $K$-Riesz basis). Finally, we give some characterizations of these sequences.