For a bounded linear operator T acting on a reproducing kernel Hilbert space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {H}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> over a nonempty set <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>, the Berezin range of T is defined by <Equation ID="Equ3"> <EquationSource Format="TEX">$ \textrm{Ber}(T)=\left\{ \langle T\hat{k}_{\lambda },\hat{k}_{\lambda }\rangle _{\mathcal {H}} : \lambda \in \Omega \right\} $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtext>Ber</mtext> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfenced close="}" open="{"> <msub> <mrow> <mo stretchy="false">⟨</mo> <mi>T</mi> <msub> <mover accent="true"> <mi>k</mi> <mo stretchy="false">^</mo> </mover> <mi>λ</mi> </msub> <mo>,</mo> <msub> <mover accent="true"> <mi>k</mi> <mo stretchy="false">^</mo> </mover> <mi>λ</mi> </msub> <mo stretchy="false">⟩</mo> </mrow> <mi mathvariant="script">H</mi> </msub> <mo>:</mo> <mi>λ</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> </mfenced> </mrow> </math></EquationSource> </Equation>and the Berezin radius is given by <Equation ID="Equ4"> <EquationSource Format="TEX">$ \textrm{ber}(T)=\sup \left\{ |\gamma | : \gamma \in \textrm{Ber}(T) \right\} , $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtext>ber</mtext> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo movablelimits="true">sup</mo> <mfenced close="}" open="{"> <mo stretchy="false">|</mo> <mi>γ</mi> <mo stretchy="false">|</mo> <mo>:</mo> <mi>γ</mi> <mo>∈</mo> <mtext>Ber</mtext> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\hat{k}_{\lambda }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mi>k</mi> <mo stretchy="false">^</mo> </mover> <mi>λ</mi> </msub> </math></EquationSource> </InlineEquation> denotes the normalized reproducing kernel at <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lambda \in \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we study the convexity of the Berezin range of finite rank operators on the Hardy space and the Bergman space over the unit disc <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">D</mi> </math></EquationSource> </InlineEquation>. We present applications of some scalar inequalities to get some operator inequalities. A characterization of closure of the numerical range of reproducing kernel Hilbert space operator in terms of convex hull of its Berezin range is also discussed.

Convexity of the Berezin range of finite rank operators

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2026年6月17日

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Athul Augustine,
M. Garayev,
P. Shankar

摘要

For a bounded linear operator T acting on a reproducing kernel Hilbert space $\mathcal {H}(\Omega )$ $H (Ω)$ over a nonempty set $\Omega $ $Ω$ , the Berezin range of T is defined by $ \textrm{Ber}(T)=\left\{ \langle T\hat{k}_{\lambda },\hat{k}_{\lambda }\rangle _{\mathcal {H}} : \lambda \in \Omega \right\} $ $Ber (T) = \{{⟨ T {\hat{k}}_{λ}, {\hat{k}}_{λ} ⟩}_{H} : λ \in Ω\}$ and the Berezin radius is given by $ \textrm{ber}(T)=\sup \left\{ |\gamma | : \gamma \in \textrm{Ber}(T) \right\} , $ $ber (T) = sup \{| γ | : γ \in Ber (T)\},$ where $\hat{k}_{\lambda }$ ${\hat{k}}_{λ}$ denotes the normalized reproducing kernel at $\lambda \in \Omega $ $λ \in Ω$ . In this paper, we study the convexity of the Berezin range of finite rank operators on the Hardy space and the Bergman space over the unit disc $\mathbb {D}$ $D$ . We present applications of some scalar inequalities to get some operator inequalities. A characterization of closure of the numerical range of reproducing kernel Hilbert space operator in terms of convex hull of its Berezin range is also discussed.