Controlled frames have been recently introduced in Hilbert spaces to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper, like the cross-Gram matrix of two different sequences which is not always a diagnostic tool, we define the controlled-Gram matrix of a sequence as a practical implement to diagnose that a given sequence is a controlled Bessel, frame or Riesz basi. Also, we discuss the cases that the operator associated to controlled Gram matrix will be bounded, invertible, Hilbert–Schmidt or a trace-class operator. Similar to standard frames, we present an explicit structure for controlled Riesz bases and show that every (U,C)-controlled Riesz basis {fk}∞ k=1 is in the form {U−1CMek}∞ k=128 , where M is a bijective operator on H. Furthermore, we propose an equivalent accessible condition to the sequence {fk}∞ k=1 30 being a (U,C)-controlled Riesz basis.
