Controlled frames have been recently introduced in Hilbert spaces to improvethe numerical efficiency of interactive algorithms for inverting the frameoperator. In this paper, unlike the cross-Gram matrix of two differentsequences which is not always a diagnostic tool, we define the controlled-Grammatrix of a sequence as a practical implement to diagnose that a given sequenceis a controlled Bessel, frame or Riesz basis. Also, we discuss the cases thatthe operator associated to controlled Gram matrix will be bounded, invertible,Hilbert-Schmidt or a trace-class operator. Similar to standard frames, wepresent an explicit structure for controlled Riesz bases and show that every$(U, C)$-controlled Riesz basis $\{f_{k}\}_{k=1}^{\infty}$ is in the form$\{U^{-1}CMe_{k}\}_{k=1}^{\infty}$, where $M$ is a bijective operator on $H$.Furthermore, we propose an equivalent accessible condition to the sequence$\{f_{k}\}_{k=1}^{\infty}$ being a $(U, C)$-controlled Riesz basis.
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