Construction of New Continuous K-g-Frames within Hilbert C∗-Modules

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In this paper, we investigate the construction of new continuous c-K-g-frames in Hilbert C*-modules, extending and generalizing existing frame theory results. Our main theorem establishes sufficient positivity conditions on an auxiliary operator T to ensure that the transformed family \(\{\Lambda_\omega T\}_{\omega \in \Omega}\) forms a continuous c-K-g-frame. Through examples, we illustrate the necessity of these positivity conditions. We also present a method for combining multiple continuous c-K-g-frames into a single continuous c-\(\sum_i K_i T_i\)-g-frame. We prove associativity of continuous c-K-g-frames under product measure spaces, and explore exactness and stability under restriction to some measurable subsets for non-decreasing continuous K-g-frames with respect to an ordered measurable space. Additionally, we characterize dual frames in this setting, providing insight into their existence and construction. Our results extend and unify various notions of continuous frames in both Hilbert spaces and Hilbert C*-modules.

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