TY - JOUR
ID - 84204
TI - Power Inequalities for the Numerical Radius of Operators in Hilbert Spaces
JO - Khayyam Journal of Mathematics
JA - KJM
LA - en
SN -
AU - Rashid, Mohammad H.M.
AD - Department of Mathematics and Statistics, Faculty of Science P.O.Box(7),
Mu’tah University, Alkarak-Jordan.
Y1 - 2019
PY - 2019
VL - 5
IS - 2
SP - 15
EP - 29
KW - Numerical Range
KW - Numerical radius
KW - Aluthge transformation
KW - strongly convex
DO - 10.22034/kjm.2019.84204
N2 - We generalize several inequalities involving powers of the numerical radius for the product of two operators acting on a Hilbert space. Moreover, we give a Jensen operator inequality for strongly convex functions. As a corollary, we improve the operator Hölder-McCarthy inequality under suitable conditions. In particular, we prove that if $f:Jrightarrow mathbb{R}$ is strongly convex with modulus $c$ and differentiable on ${rm int}(J)$ whose derivative is continuous on ${rm int}(J)$ and if $T$ is a self-adjoint operator on the Hilbert space $cal{H}$ with $sigma(T)subset {rm int}(J)$, then $$langle T^2x,xrangle-langle Tx,xrangle^2leq dfrac{1}{2c}(langle f'(T)Tx,xrangle -langle Tx,xrangle langle f'(T)x,xrangle)$$ for each $xincal{H}$, with $|x|=1$.
UR - http://www.kjm-math.org/article_84204.html
L1 - http://www.kjm-math.org/article_84204_a321253ff5f81d65f8472735f8eb5f80.pdf
ER -