Continuous g‐frame and g‐Riesz sequences in Hilbert spaces

A continuous g‐frame is a generalization of g‐frames and continuous frames, but they behave much differently from g‐frames due to the underlying characteristic of measure spaces. Now, continuous g‐frames have been extensively studied, while continuous g‐sequences such as continuous g‐frame sequence, g‐Riesz sequences, and continuous g‐orthonormal systems have not. This paper addresses continuous g‐sequences. It is a continuation of Zhang and Li, in Numer. Func. Anal. Opt., 40 (2019), 1268‐1290, where they dealt with g‐sequences. In terms of synthesis and Gram operator methods, we in this paper characterize continuous g‐Bessel, g‐frame, and g‐Riesz sequences, respectively, and obtain the Pythagorean theorem for continuous g‐orthonormal systems. It is worth that our results are similar to the case of g‐ones, but their proofs are nontrivial. It is because the definition of continuous g‐sequences is different from that of g‐sequences due to it involving general measure space.