Generalized Dual Hilbert–Schmidt Frames and Their Topological Properties

This paper addresses the Hilbert–Schmidt frame (HS-frame) theory. We introduce the concept of generalized dual HS-frame (g-dual HS-frame) which generalizes that of g-dual frame. We prove that two equivalent HS-frames form a g-dual HS-frame pair, characterize operators on \(\ell ^2\) ${\ell }^2$ that transform a pair of HS-Riesz bases into a g-dual HS-frame pair, and present a parametric expression of all g-dual HS-frames of an arbitrarily given HS-frame. Also the perturbation-stability and topological properties of g-dual HS-frames are investigated. Finally, applying our results, we not only recover some known results but also derive some new results in the classical Hilbert space frame setting.