Operator Inequalities and Several Characterizations of the $\lambda$-Mean Transform

We broaden Buzano-type inequalities to provide novel numerical radius bounds for operators of the type $AXB$, thereby generalizing the results obtained by Sababheh et al. For the $\lambda$-mean transform $M_{\lambda }(T)$, we provide a counterexample demonstrating that $r_{\sigma }(M_{\lambda }(T))\le r_{\sigma }(T)$ fails to hold in general for $\lambda \in (0,1)$, establish that $(r_{\omega }(M_{\lambda }(T)){)}^n$ and $r_{\omega }(T^n)$ are typically incomparable for $n\ge 2$, and confirm that $M_{\lambda }(T^{\ast })=(M_{\lambda }(T){)}^{\ast }$ is valid for $\lambda \in [0,1)$ if and only if $T$ is a member of a newly established $\sigma$-class. Furthermore, we examine the transformation characteristics of $T$ and the tensor products $T\otimes S$, refine Zamani’s inequalities, and unify operator modulus bounds $\mathrm{\mid }\tilde{T}\mathrm{\mid }\le \mathrm{\mid }\widehat{T}\mathrm{\mid }\le \mathrm{\mid }T\mathrm{\mid }$. In this application, we demonstrate that the conditions for norm preservation, $\mathrm{\parallel }\tilde{T}\mathrm{\parallel }=\mathrm{\parallel }T\mathrm{\parallel }$ and $\mathrm{\parallel }{M}_{\lambda }(T)\mathrm{\parallel }=\mathrm{\parallel }T\mathrm{\parallel }$, are equivalent to the statement $\mathrm{\parallel }{T}^2\mathrm{\parallel }=\mathrm{\parallel }T{\mathrm{\parallel }}^2$, and we offer precise norm estimates for $2\times 2$ off-diagonal block operator matrices under $\lambda$-mean transformation.