Harmonic Analysis and Applications

June 4-8, 2018


"Gram and cross-Gram operators of continuous frames in Hilbert spaces"

Rahimi, Asghar

One of the frame related matrices or operators is the Gram matrix or operator. Unlike, other related operators ( analysis, synthesis and frame operator which their domain, range or both of them are in underlying Hilbert spaces), this operator's domain and range lies in the representation sequence space $\ell^2$ ( in discrete case) or the function space $L^2$ ( in continuous case), which makes it useful and sometimes hard to work. In this manuscript, we review and study continuous frames in point of view of this operator. Then, for two given Bessel mappings $E$ and $F$, we define the so called cross-Gram operator $G_{E,F}$. We show that in case $F$ and $G$ are Bessel mappings then $G_{E,F}$ is bounded. Also, we show that $G_{E,F}$ is bounded and invertible if and only if both $E$ and $F$ are Riesz type.

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