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Generalized frames for controlled operators in Hilbert spaces. (English) Zbl 1428.42062

Let \(H\) be a Hilbert space. A countable family of bounded Hilbert space operators \(\Lambda =(\Lambda_i)\), \(\Lambda_i :H \rightarrow H_i, i\in I\), is said to be a \((T,U)\)-controlled \(g\)-frame for \(H\) if \(\Lambda\) is a \(g\)-Bessel sequence and, in addition, if there exist bounded invertible operators \(T\), \(U\) on \(H\) and positive constants \(A\) and \(B\) such that \[ A\|x\|^2\leq \sum_{i\in I}\langle \Lambda_iTx,\Lambda_iUx\rangle \leq B\|x\|^2,\,\,\forall x\in H. \] The paper contains some elementary properties of such families of operators.

MSC:

42C15 General harmonic expansions, frames
47B99 Special classes of linear operators
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