In this article, we improve some Berezin number inequalities concerning a Hilbert space. It is shown that if T is a bounded linear operator on a Hilbert space, then for any r≥1{r\geq 1}, 𝐛𝐞𝐫2r(T)≤12𝐛𝐞𝐫r(|T*|2(1-t)|T|2t)+14∥|T|4rt+|T*|4r(1-t)∥𝐛𝐞𝐫(0≤t≤1),\mathbf{ber}^{2r}(T)\leq\frac{1}{2}\mathbf{ber}^{r}({{|{{T}^{*}}|}^{2(1-t)}}{{% |T|}^{2t}})+\frac{1}{4}{{\|{{|T|}^{4rt}}+{{|{{T}^{*}}|}^{4r(1-t)}}\|}_{\mathbf% {ber}}}\quad(0\leq t\leq 1), where |T|=(T*T)12{|T|={{({{T}^{*}}T)}^{\frac{1}{2}}}}.
在本文中，我们改进了一些关于希尔伯特空间的贝雷津数不等式。结果表明，如果 T 是希尔伯特空间上的有界线性算子，那么对于任意 r ≥ 1 {r\geq 1} ，𝐛𝐞𝐫𝐫是有界线性算子。 𝐛𝐞𝐫 2 r ( T ) ≤ 1 2 𝐛𝐞𝐫 r ( | T * | 2 ( 1 - t ) | T | 2 t ) + 1 4 ∥ | T | 4 r t + | T * | 4 r ( 1 - t ) ∥ 𝐛𝐞𝐫 ( 0 ≤ t ≤ 1 ) 、 \mathbf{ber}^{2r}(T)\leq\frac{1}{2}\mathbf{ber}^{r}({{|{{T}^{*}}|}^{2(1-t)}}{{% |T|}^{2t}})+\frac{1}{4}{{\|{{|T|}^{4rt}}+{{|{{T}^{*}}|}^{4r(1-t)}}\|}_{\mathbf% {ber}}}\quad(0\leq t\leq 1), 其中 | T | = ( T * T ) 1 2 {|T|={{({{T}^{*}}T)}^{\frac{1}{2}}}} .
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The Georgian Mathematical Journal was founded by the Georgian National Academy of Sciences and A. Razmadze Mathematical Institute, and is jointly produced with De Gruyter. The concern of this international journal is the publication of research articles of best scientific standard in pure and applied mathematics. Special emphasis is put on the presentation of results obtained by Georgian mathematicians.
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