LAUSR

Abstract For a bounded linear operator T on a complex Hilbert space ℋ {\mathcal{H}} , we define the weighted real and imaginary parts of the operator T as ℜ t ⁢ ( T ) := ( 1 - t ) ⁢ T ∗ + t ⁢ T {\mathfrak{R}_{t}(T):=(1-t)T^{\ast}+tT} and ℑ t ⁢ ( T ) := ( 1 - t ) ⁢ T - t ⁢ T ∗ i {\mathfrak{I}_{t}(T):=\frac{(1-t)T-tT^{\ast}}{i}} for t ∈ [ 0 , 1 ] {t\in[0,1]} , where T ∗ {T^{\ast}} denotes the adjoint operator of T. By setting T t := ℜ t ⁢ ( T ) + i ⁢ ℑ t ⁢ ( T ) {T_{t}:=\mathfrak{R}_{t}(T)+i\mathfrak{I}_{t}(T)} , we define the weighted norm ∥ T ∥ t := ∥ T t ∥ {\|T\|_{t}:=\|T_{t}\|} . Another weighted numerical radius is given by w ~ t ⁢ ( T ) = sup ⁡ { ∥ ℜ t ⁢ ( e i ⁢ φ ⁢ T ) ∥ : φ ∈ ℝ } {\widetilde{w}_{t}(T)=\sup\{\|\mathfrak{R}_{t}(e^{i\varphi}T)\|:\varphi\in% \mathbb{R}\}} for t ∈ [ 0 , 1 ] {t\in[0,1]} . In this paper, we present new power inequalities for the weighted operator norm ∥ ⋅ ∥ t {\|\cdot\|_{t}} and the weighted numerical radius w ~ t ⁢ ( ⋅ ) {\widetilde{w}_{t}(\,\cdot\,)} . Additionally, we discuss the special cases of the classical numerical radius and norm of Hilbert space operators.