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.