Sum of Disjoint Frame Sequences
We show that, if A and B are bounded operators on a Hilbert space and X and Y are strongly disjoint (orthogonal) frame sequences, then A ( X ) + B ( Y ) is a frame sequence if and only if the sum of the ranges of the synthesis operators of A ( X ) and B ( Y ) is closed. We also show that, given two...
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| Veröffentlicht in: | Bulletin of the Malaysian Mathematical Sciences Society 2020-01, Vol.43 (1), p.321-331 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | We show that, if
A
and
B
are bounded operators on a Hilbert space and
X
and
Y
are strongly disjoint (orthogonal) frame sequences, then
A
(
X
)
+
B
(
Y
)
is a frame sequence if and only if the sum of the ranges of the synthesis operators of
A
(
X
) and
B
(
Y
) is closed. We also show that, given two disjoint frame sequences, the sum is a frame sequence if the sum of the ranges of the synthesis operators is closed but not vice versa. A counterexample is given by a couple of frames of shifts for two finitely generated shift-invariant spaces of
L
2
(
R
)
. |
|---|---|
| ISSN: | 0126-6705 2180-4206 |
| DOI: | 10.1007/s40840-018-0682-1 |