Power inequalities for weighted numerical radius and norm of operators

Abstract For a bounded linear operator T on a complex Hilbert space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℋ</m:mi> </m:math> {\mathcal{H}} , we define the weighted real and imaginary parts of the operator T as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>ℜ</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>T</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>:=</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>-</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:msup> <m:mi>T</m:mi> <m:mo>∗</m:mo> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>⁢</m:mo> <m:mi>T</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:math> {\mathfrak{R}_{t}(T):=(1-t)T^{\ast}+tT} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>ℑ</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>T</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>:=</m:mo> <m:mfrac> <m:mrow> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>-</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>T</m:mi> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mi>T</m:mi> <m:mo>∗</m:mo> </m:msup> </m:mrow> </m:mrow> <m:mi>i</m:mi> </m:mfrac> </m:mrow> </m:math> {\mathfrak{I}_{t}(T):=\frac{(1-t)T-tT^{\ast}}{i}} for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>t</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:math> {t\in[0,1]} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>T</m:mi> <m:mo>∗</m:mo> </m:msup> </m:math> {T^{\ast}} denotes the adjoint operator of T . By setting <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>T</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo>:=</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>ℜ</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>T</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>ℑ</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>T</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> {T_{t}:=\mathfrak{R}_{t}(T)+i\mathfrak{I}_{t}(T)} , we define the weighted norm <jats:alternative