Numerical radius inequalities of bounded linear operators and (α,β)-normal operators

We obtain various upper bounds for the numerical radius w(T) of a bounded linear operator T defined on a complex Hilbert space H, by developing the upper bounds for the alpha-norm of T, which is defined as & Vert;T & Vert;(alpha)= sup{root alpha|T alpha x,x|(2)+(1-alpha)& Vert;Tx & Vert;2:x is an element of H,& Vert;x & Vert;=1} for 0 <= alpha <= 1 . Further, we prove that w(T)<=root(& Vert;alpha|T|+(1-alpha)|T & lowast;|& Vert;)& Vert;T & Vert;<=& Vert;T & Vert;,for all alpha is an element of[0,1].For 0 <=alpha <= 1 <=beta,the operatorTis called (alpha, beta)-normal if alpha 2T & lowast;T <= TT & lowast;<=beta 2T & lowast;Tholds. Note that every invertible operator is an (alpha, beta)-normal operator for suitable values of alpha and beta. Among other lower bounds for the numerical radius of an (alpha, beta)-normal operator T, we show that w(T) >=root max{1+alpha 2,1+1 beta 2}& Vert;T & Vert;24+|& Vert; (T)& Vert;2-& Vert; (T)& Vert;2|2 >= max {root 1+alpha(2),root 1+1 beta(2)}& Vert;T & Vert;/2>& Vert;T & Vert;/2, where(T)and(T) are the real part and imaginary part ofT,respectively.