Perturbation and Stability of Dual and Woven Hilbert-Schmidt Frames

Hilbert-Schmidt (HS) frames are the p = ۲ instance of von Neumann-Schatten operator-valued frames, with coefficients in the Hilbert space C۲(H). We develop an operator-norm perturbation theory for duality and weaving in the HS-frame setting and derive explicit, verifiable robustness margins in terms of synthesis-operator deviations. First, we provide a computable Neumann-series criterion for the invertibility of the perturbed HS-frame operator, together with norm bounds for S¹ and S¹-S¹. Second, we establish a Lipschitz-type stability estimate for the canonical dual synthesis operator whose bound vanishes as the perturbation level tends to zero. Third, we prove a simultaneous perturbation theorem for m-woven HS-frames, yielding universal weaving bounds with explicit dependence on the perturbation radii.

کلیدواژه ها:

Hilbert-Schmidt frame ، canonical dual ، perturbation ، weaving frames ، Neumann series