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<p>The aim of this work is to study frame theory in quaternionic Hilbert spaces. We provide a characterization of frames in these spaces through the associated operators. Additionally, we examine frames of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\{L(u_i)\}_{i \in I}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <mi>L</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, where <i>L</i> is a bounded right <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {H}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">H</mi> </math></EquationSource> </InlineEquation>-linear operator and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\{u_i\}_{i \in I}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>u</mi> <mi>i</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is a frame. We also show that every bounded, positive, invertible right <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {H}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">H</mi> </math></EquationSource> </InlineEquation>-linear operator arises as a frame operator.</p>

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Frames and operators on quaternionic Hilbert spaces

  • 期刊论文

  • 发表日期：

    2026年3月9日

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  • Najib Khachiaa

摘要

The aim of this work is to study frame theory in quaternionic Hilbert spaces. We provide a characterization of frames in these spaces through the associated operators. Additionally, we examine frames of the form \(\{L(u_i)\}_{i \in I}\) { L ( u i ) } i ∈ I , where L is a bounded right \(\mathbb {H}\) H -linear operator and \(\{u_i\}_{i \in I}\) { u i } i ∈ I is a frame. We also show that every bounded, positive, invertible right \(\mathbb {H}\) H -linear operator arises as a frame operator.