Construction of continuous controlled \(K\text{-}g\)-fusion frames in Hilbert spaces (Q6585999)

In this work, continuous controlled \(K\text{-}g\)-fusion frames in Hilbert spaces are explored, establishing various properties. It is shown that under certain conditions, any continuous controlled \(K\text{-}g\)-fusion frame is equivalent to a continuous \(K\text{-}g\)-fusion frame. Additionally, a necessary and sufficient condition is established for a continuous controlled \(g\)-fusion Bessel family to be a continuous controlled \(K\text{-}g\)-fusion frame using a quotient operator. The paper also investigates stability results for continuous controlled \(g\)-fusion frames.\N\NThe study considers \(H\) as a separable Hilbert space with an associated inner product \(\langle\cdot, \cdot\rangle\). The collection of all closed subspaces of \(H\) is denoted by \(\mathbb{H}\), and \(I_H\) represents the identity operator on \(H\). \(\mathcal{B}(H)\) denotes the space of all bounded linear operators on \(H\), with \( \mathcal{N}(S)\) and \(\mathcal{R}(S)\) representing the null space and the range of \(S\), respectively. \(P_M\) denotes the orthonormal projection onto a closed subspace \(M \subset H\). The set \(\mathcal{G} \mathcal{B}(H)\) includes all bounded linear operators with bounded inverses, and \(\mathcal{G} \mathcal{B}^{+}(H)\) represents positive operators within this set.\N\NA continuous version of a controlled \(K\text{-}g\)-fusion frame for \(H\) is presented, expanding on recent results for controlled \(K\text{-}g\)-fusion frames. A continuous \((T, U)\)-controlled \(K\text{-}g\)-fusion frame for \(H\) is defined as a family \(\Lambda_{T U}=\{(F(x), \Lambda_x, v(x))\}_{x \in X}\) that satisfies specific conditions involving projections \(P_{F(x)}\) and frame bounds \(A\) and \(B\). Different cases arise when specific choices are made for the operators \(T\), \(U\), and \(K\).\N\NThe paper continues by discussing various properties of these frames, including the construction of new types of continuous controlled \(g\)-fusion frames from existing ones, and the equivalence of continuous controlled \(K\text{-}g\)-fusion frames to continuous controlled \(g\)-fusion frames under certain conditions. Stability of these frames under perturbation is also considered, with a focus on ensuring that the frame remains stable when slight changes occur in the system.