Classification of general sequences by frame-related operators (Q2890686)

A sequence \(\Psi=\{\psi_j\}\) in a Hilbert space \(H\) can be a generalization of a basis in many possible ways. These are best known in the context of frames and their related operators.NEWLINENEWLINE In this paper all these relations and possible generalizations are surveyed starting from arbitrary sequences. First the definitions are recalled of \(\Psi\) being an X sequence where X can be Bessel, lower frame, frame, Riesz basis, or Riesz-Fisher. These are related to different boundedness conditions from above and from below and convergence of the sequence of coordinates \(\{\langle f,\psi_j\rangle\}\) that can be associated with an \(f\in H\).NEWLINENEWLINEThe related operators are the analysis operator (\(f\to \{\langle f,\psi_j\rangle\}\)), the synthesis operator (\(c\to \sum_j c_j\psi_j\)), the frame operator (\(f\to \sum_j\langle f,\psi_j\rangle\psi_j\)) and the Gram operator (\(c\to \{\sum_l\langle \psi_k,\psi_l\rangle c_l\}\)).NEWLINENEWLINE For a general sequence \(\Psi\), these operators can be unbounded. A careful analysis of the domains, ranges, and kernels of the operators is given. These properties are used to characterize the sequence to be one of the X sequences and vice versa.NEWLINENEWLINE The different X sequences are also considered as transforms of an orthonormal basis and they are characterized by the properties of this transformation operator.