New estimates for the Berezin number of Hilbert space operators

Abstract

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} , 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), where | T | = ( T * ⁢ T ) 1 2 {|T|={{({{T}^{*}}T)}^{\frac{1}{2}}}} .