Operator Inequalities and Several Characterizations of the $λ$-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 $λ$-mean transform $M_λ(T)$, we provide a counterexample demonstrating that $r_σ(M_λ(T)) \le r_σ(T)$ fails to hold in general for $λ\in (0, 1)$, establish that $(r_ω(M_λ(T)))^n$ and $r_ω(T^n)$ are typically incomparable for $n \ge 2$, and confirm that $M_λ(T^*) = (M_λ(T))^*$ is valid for $λ\in [0, 1)$ if and only if $T$ is a member of a newly established $σ$-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 $|\widetilde{T}| \le |\widehat{T}| \le |T|$. In this application, we demonstrate that the conditions for norm preservation, $\|\widetilde{T}\| = \|T\|$ and $\|M_λ(T)\| = \|T\|$, are equivalent to the statement $\|T^2\| = \|T\|^2$, and we offer precise norm estimates for $2 \times 2$ off-diagonal block operator matrices under $λ$-mean transformation.