Some constructions of \(K\)-frames and their duals
Let \(H\) be a separable Hilbert space and \(K\) a bounded operator on \(H.\) Then a sequence \(F:=\left\{f_i\right\}_{i\in I}\subset H\) is called a \(K\)-frame for \(H\) if there exist constants \(A, B > 0\) such that \[ A||K^\ast f||^2\leq \sum_{i\in I}|< f, f_i >|^2 \leq B||f||^2,\;f\in H. \] \(K\)-frames were introduced in [\textit{L. Găvruţa}, Appl. Comput. Harmon. Anal. 32, No. 1, 139--144 (2012; Zbl 1230.42038)] to study atomic systems with respect to a bounded operator. They are a generalization of ordinary frames. In fact, every frame sequence \(F:=\left\{f_i\right\}_{i\in I}\) can be considered as a \(K\)-frame where \(K\) is the orthogonal projection onto the closed subspace generated by \(F.\) On the other hand many properties of ordinary frames do not hold for \(K\)-frames. The present paper introduces the notion of \(K\)-dual of a \(K\)-frame and investigates its properties. A Bessel sequence \(\left\{g_i\right\}_{i\in I}\) is said to be a \(K\)-dual of the \(K\)-frame \(\left\{f_i\right\}_{i\in I}\) if \[ Kf = \sum_{i\in I} < f, g_i > f_i,\;f\in H. \] In particular, the so-called canonical \(K\)-dual is discussed and some characterizations of \(K\)-duals are presented. The final section contains some approaches for the construction of \(K\)-frames and their \(K\)-duals. In this respect, the authors first introduce the notion of approximate \(K\)-dual.
scientific article; zbMATH DE number 6816569