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Peer-reviewedCC BY 4.0Mathematics

Multipliers of Continuous K-Frames

  • Khadija Mabrouk1
  • Maryam Gharamah Alshehri2
  • Mohamed Rossafi1
  1. 1Laboratory Analysis, Geometry and Applications, University of Ibn Tofail, Kenitra 14000, Morocco
  2. 2Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
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Abstract

We study multipliers Mm,f,g of continuous K-frames over a σ-finite measure space. We give criteria for invertibility on Ran(K) governed by ∥m−1∥∞, and show that previously published criteria stated in terms of ∥m∥∞ have hypotheses that no admissible data can satisfy, so that they are vacuous rather than false. A Douglas-type characterization replaces the known one: Mm,f,g is invertible on Ran(K) exactly when Ran(K)⊆Ran(πRan(K)Mm¯,g,f). We then characterize the existence of continuous K-right and K-left inverses by range inclusions, identify those of minimal norm, and represent one-sided inverses as multipliers of continuous K-duals; a two-dimensional example shows that a representation formula for K-left inverses previously stated in the discrete setting is not correct as stated. Finally, we prove that pointwise biorthogonality is ineffective when μ has no atoms, and replace it by a μ-weighted biorthogonality along a partition of Ω into blocks, which we then extend to families that are not constant on the blocks.

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Cite

BibTeXRIS
BibTeX
@article{mabrouk2026multipliers,
  title = {{Multipliers of Continuous K-Frames}},
  author = {Khadija Mabrouk and Maryam Gharamah Alshehri and Mohamed Rossafi},
  journal = {Mathematics},
  year = {2026},
  month = oct,
  doi = {10.3390/math14203794},
  url = {https://www.mdpi.com/2227-7390/14/20/3794},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Multipliers of Continuous K-Frames
AU  - Khadija Mabrouk
AU  - Maryam Gharamah Alshehri
AU  - Mohamed Rossafi
PY  - 2026
DA  - 2026/10/19
JO  - Mathematics
DO  - 10.3390/math14203794
UR  - https://www.mdpi.com/2227-7390/14/20/3794
AB  - We study multipliers Mm,f,g of continuous K-frames over a σ-finite measure space. We give criteria for invertibility on Ran(K) governed by ∥m−1∥∞, and show that previously published criteria stated in terms of ∥m∥∞ have hypotheses that no admissible data can satisfy, so that they are vacuous rather than false. A Douglas-type characterization replaces the known one: Mm,f,g is invertible on Ran(K) exactly when Ran(K)⊆Ran(πRan(K)Mm¯,g,f). We then characterize the existence of continuous K-right and K-left inverses by range inclusions, identify those of minimal norm, and represent one-sided inverses as multipliers of continuous K-duals; a two-dimensional example shows that a representation formula for K-left inverses previously stated in the discrete setting is not correct as stated. Finally, we prove that pointwise biorthogonality is ineffective when μ has no atoms, and replace it by a μ-weighted biorthogonality along a partition of Ω into blocks, which we then extend to families that are not constant on the blocks.
SN  - 2227-7390
ER  -