Consider a non-zero positive operator A on a complex Hilbert space <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">H</mi> <mo>,</mo> <mo stretchy="false">〈</mo> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">〉</mo> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(\mathfrak{H}, \langle \cdot , \cdot \rangle )$</EquationSource> </InlineEquation>. This operator induces an A-semi-inner product defined by <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">∣</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="bold">A</mi> </msub> <mo>:</mo> <mo>=</mo> <mo stretchy="false">〈</mo> <mi mathvariant="bold">A</mi> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">〉</mo> </math></EquationSource> <EquationSource Format="TEX">$(u \mid v)_{\mathbf{A}} := \langle \mathbf{A}u, v \rangle $</EquationSource> </InlineEquation>. The space <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">H</mi> <mo>,</mo> <msub> <mrow> <mo stretchy="false">∥</mo> <mo>⋅</mo> <mo stretchy="false">∥</mo> </mrow> <mi mathvariant="bold">A</mi> </msub> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(\mathfrak{H}, \|\cdot \|_{\mathbf{A}})$</EquationSource> </InlineEquation> then becomes a semi-Hilbert space, where <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">∥</mo> <mo>⋅</mo> <mo stretchy="false">∥</mo> </mrow> <mi mathvariant="bold">A</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$\|\cdot \|_{\mathbf{A}}$</EquationSource> </InlineEquation> is the seminorm generated by this A-semi-inner product. The primary focus of this work is to establish novel additive bounds for Bessel’s inequality within the framework of semi-Hilbert spaces. Furthermore, we explore applications of these new bounds to several A-seminorms that are associated with n-tuples of operators.

Bessel-type inequalities in semi-Hilbert spaces with applications to operator tuples

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2026年3月12日

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Feryal Aladsani,
Asmahan Alajyan,
Silvestru Sever Dragomir,
Kais Feki

摘要

Consider a non-zero positive operator A on a complex Hilbert space $(H, 〈 \cdot, \cdot 〉)$ $(\mathfrak{H}, \langle \cdot , \cdot \rangle )$ . This operator induces an A-semi-inner product defined by ${(u ∣ v)}_{A} : = 〈 A u, v 〉$ $(u \mid v)_{\mathbf{A}} := \langle \mathbf{A}u, v \rangle $ . The space $(H, {∥ \cdot ∥}_{A})$ $(\mathfrak{H}, \|\cdot \|_{\mathbf{A}})$ then becomes a semi-Hilbert space, where ${∥ \cdot ∥}_{A}$ $\|\cdot \|_{\mathbf{A}}$ is the seminorm generated by this A-semi-inner product. The primary focus of this work is to establish novel additive bounds for Bessel’s inequality within the framework of semi-Hilbert spaces. Furthermore, we explore applications of these new bounds to several A-seminorms that are associated with n-tuples of operators.