Duality principles in Hilbert-Schmidt frame theory. (English) Zbl 1475.42043
The concept of frame has several generalizations such as pseudo-frame, fusion frame (or frame of subspaces) and generalized frame (known as g-frame), among others. In particular, Hilbert-Schmidt (HS) frames are sequences of HS operators and are a more general version of g-frames. In this context, the notions of HS-Bessel sequences, HS-Riesz bases and HS-orthonormal bases are defined.
In the present paper, the notion of HS-R-dual sequence is introduced and used to establish some results about duality. It is proved that a sequence is an HS-frame (HS-frame sequence, HS-Riesz basis) if and only if its HS-R-dual sequence is an HS-Riesz sequence (HS-frame sequence, HS-Riesz basis) and the (unitary) equivalence between two HS-frames is characterized in terms of their HS-R-duals and transition matrices, respectively. HS-Rduals are characterized and it is proved that, given an HS-frame, among all its dual HS-frames, only the canonical dual admits minimal-norm HS-R-dual. Using HS-R-duals, dual HS-frame pairs are also characterized.
In the present paper, the notion of HS-R-dual sequence is introduced and used to establish some results about duality. It is proved that a sequence is an HS-frame (HS-frame sequence, HS-Riesz basis) if and only if its HS-R-dual sequence is an HS-Riesz sequence (HS-frame sequence, HS-Riesz basis) and the (unitary) equivalence between two HS-frames is characterized in terms of their HS-R-duals and transition matrices, respectively. HS-Rduals are characterized and it is proved that, given an HS-frame, among all its dual HS-frames, only the canonical dual admits minimal-norm HS-R-dual. Using HS-R-duals, dual HS-frame pairs are also characterized.
Reviewer: Patricia Mariela Morillas (San Luis)
MSC:
| 42C15 | General harmonic expansions, frames |
| 46C50 | Generalizations of inner products (semi-inner products, partial inner products, etc.) |
| 47A58 | Linear operator approximation theory |
| 41A35 | Approximation by operators (in particular, by integral operators) |
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