In this paper, we define and study g-Riesz bases and g-orthonormal bases for a right quaternionic Hilbert space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {H}^R(\mathfrak {Q})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mi>R</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">Q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with respect to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\{\mathbb {H}^R_i(\mathfrak {Q})\}_{i\in \mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msubsup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mi>i</mi> <mi>R</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">Q</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>i</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> and provide necessary and sufficient conditions for their characterization using the sequence induced by a family of right bounded linear operators. A characterization of dual g-frames in terms of the induced sequence is established. It is shown that every g-Riesz basis for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {H}^R(\mathfrak {Q})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mi>R</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">Q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with respect to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\{\mathbb {H}^R_i(\mathfrak {Q})\}_{i\in \mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msubsup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mi>i</mi> <mi>R</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">Q</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>i</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> can be realized as the image of a g-orthonormal basis under a bounded invertible linear operator. We also introduce the notion of alternate dual g-frames and present a characterization of dual g-frames. Finally, approximate dual g-frames are defined, and several characterizations of approximate dual g-frames are obtained.

ON DUALS OF $g-$FRAMES IN QUATERNIONIC HILBERT SPACES

期刊论文
发表日期：

2026年4月14日

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S. K. Sharma,
Nikhil Khanna,
S. K. Kaushik

摘要

In this paper, we define and study g-Riesz bases and g-orthonormal bases for a right quaternionic Hilbert space $\mathbb {H}^R(\mathfrak {Q})$ $H^{R} (Q)$ with respect to $\{\mathbb {H}^R_i(\mathfrak {Q})\}_{i\in \mathbb {N}}$ ${H_{i}^{R} (Q)}_{i \in N}$ and provide necessary and sufficient conditions for their characterization using the sequence induced by a family of right bounded linear operators. A characterization of dual g-frames in terms of the induced sequence is established. It is shown that every g-Riesz basis for $\mathbb {H}^R(\mathfrak {Q})$ $H^{R} (Q)$ with respect to $\{\mathbb {H}^R_i(\mathfrak {Q})\}_{i\in \mathbb {N}}$ ${H_{i}^{R} (Q)}_{i \in N}$ can be realized as the image of a g-orthonormal basis under a bounded invertible linear operator. We also introduce the notion of alternate dual g-frames and present a characterization of dual g-frames. Finally, approximate dual g-frames are defined, and several characterizations of approximate dual g-frames are obtained.

ON DUALS OF \(g-\)FRAMES IN QUATERNIONIC HILBERT SPACES

摘要