<p>This paper investigates the dilation problem for oblique dual pairs of g-frame sequences in Hilbert spaces. It is demonstrated that any oblique dual pair (Type I dual pair) of g-frame sequences in a Hilbert space can be dilated to an oblique dual pair (Type I dual pair) of g-Riesz sequences in a larger Hilbert space. Furthermore, a characterization is established for a g-frame sequence to possess a Parseval Type II dual through orthogonal dilation. Finally, a condition is provided under which an oblique dual pair of g-frame sequences in a Hilbert space can be obliquely dilated to a dual pair of g-Riesz bases for the same space.</p>

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Dilations of oblique dual pairs of g-frame sequences

  • Xiujiao Chi,
  • Pengtong Li,
  • Yangyang Shi

摘要

This paper investigates the dilation problem for oblique dual pairs of g-frame sequences in Hilbert spaces. It is demonstrated that any oblique dual pair (Type I dual pair) of g-frame sequences in a Hilbert space can be dilated to an oblique dual pair (Type I dual pair) of g-Riesz sequences in a larger Hilbert space. Furthermore, a characterization is established for a g-frame sequence to possess a Parseval Type II dual through orthogonal dilation. Finally, a condition is provided under which an oblique dual pair of g-frame sequences in a Hilbert space can be obliquely dilated to a dual pair of g-Riesz bases for the same space.