Complex Analysis and Operator Theory ( IF 0.7 ) Pub Date : 2021-01-27 , DOI: 10.1007/s11785-021-01078-7 M. T. Garayev , M. W. Alomari
For a bounded linear operator, acting in the reproducing kernel Hilbert space \({\mathcal {H}}={\mathcal {H}}\left( \Omega \right) \) over some set \(\Omega \), its Berezin symbol (or Berezin transform)\(\widetilde{\text { }A}\) is defined by
$$\begin{aligned} {\widetilde{A}}\left( \lambda \right) :=\left\langle A{\widehat{k}}_{\lambda },{\widehat{k}}_{\lambda }\right\rangle ,\text { }\lambda \in \Omega , \end{aligned}$$which is a bounded complex-valued function on \(\Omega ;\) here \({\widehat{k}}_{\lambda }:=\frac{{\widehat{k}}_{\lambda }}{\left\| k_{\lambda }\right\| _{{\mathcal {H}}}}\) is the normalized reproducing kernel of \({\mathcal {H}}\). The Berezin set and the Berezin number of an operator A are defined respectively by
$$\begin{aligned} \mathrm {Ber}\left( A\right) :=\mathrm {Range}\left( {\widetilde{A}}\right) =\left\{ {\widetilde{A}}\left( \lambda \right) :\lambda \in \Omega \right\} \end{aligned}$$and
$$\begin{aligned} \mathrm {ber}\left( A\right) :=\sup \left\{ \left| \gamma \right| :\gamma \in \mathrm {Ber}\left( A\right) \right\} =\sup _{\lambda \in \Omega }\left| {\widetilde{A}}\left( \lambda \right) \right| . \end{aligned}$$Since \(\mathrm {Ber}\left( A\right) \subset W\left( A\right) \) (numerical range) and \(\mathrm {ber}\left( A\right) \le w\left( A\right) \) (numerical radius), it is natural to investigate these new numerical quantities of operators and to get some similar results as for numerical range and numerical radius. In this paper, we prove many different type inequalities, including power inequality \(\mathrm {ber}\left( A^{n}\right) \le \mathrm {ber}\left( A\right) ^{n}\) for the Berezin number of operators. We also study the uncertainty principle for Berezin symbols and we describe spectrum and compactness of functions of model operator in terms of Berezin symbols. Some related problems for de Branges-Rovnyak space operators are also discussed.
中文翻译：
Berezin运算符数量不等式及相关问题
对于有界线性算子，在某些集合\（\ Omega \）上，在再生内核希尔伯特空间\（{\ mathcal {H}} = {\ mathcal {H}} \ left（\ Omega \ right）\）中起作用，其Berezin符号（或Berezin变换）\（\ widetilde {\ text {} A} \）由
$$ \ begin {aligned} {\ widetilde {A}} \ left（\ lambda \ right）：= \ left \ langle A {\ widehat {k}} _ {\ lambda}，{\ widehat {k}} _ {\ lambda} \ right \ rangle，\ text {} \ lambda \ in \ Omega，\ end {aligned} $$这是\（\ Omega; \）在这里\（{\ widehat {k}} _ {\ lambda}：= \ frac {{\ widehat {k}} _ {\ lambda}} { \ left \ | k _ {\ lambda} \ right \ | _ {{\ mathcal {H}}}} \\）是\（{\ mathcal {H}} \}的规范化复制内核。算子A的Berezin集和Berezin数分别定义为
$$ \ begin {aligned} \ mathrm {Ber} \ left（A \ right）：= \ mathrm {Range} \ left（{\ widetilde {A}} \ right）= \ left \ {{\ widetilde {A} } \ left（\ lambda \ right）：\ lambda \ in \ Omega \ right \} \ end {aligned} $$和
$$ \ begin {aligned} \ mathrm {ber} \ left（A \ right）：= \ sup \ left \ {\ left | \ gamma \ right | ：\ gamma \ in \ mathrm {Ber} \ left（A \ right）\ right \} = \ sup _ {\ lambda \ in \ Omega} \ left | {\ widetilde {A}} \ left（\ lambda \ right）\ right | 。\ end {aligned} $$由于\（\ mathrm {Ber} \ left（A \ right）\ subset W \ left（A \ right）\）（数值范围）和\（\ mathrm {ber} \ left（A \ right）\ le w \ left（A \ right）\）（数值半径），很自然地研究这些新的算子数值并获得与数值范围和数值半径相似的结果。在本文中，我们证明了许多不同的类型不等式，包括幂不等式\（\ mathrm {ber} \ left（A ^ {n} \ right）\ le \ mathrm {ber} \ left（A \ right）^ {n} \）代表Berezin运算符的数量。我们还研究了Berezin符号的不确定性原理，并根据Berezin符号描述了模型算子函数的频谱和紧凑性。还讨论了de Branges-Rovnyak空间算子的一些相关问题。
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