Generalized Cartesian decomposition and numerical radius inequalities

Let T = {lambda is an element of C :| lambda |= 1}. Every linear operator T on a complex Hilbert space H can be decomposed as T = T + lambda T*/2 + i T - lambda T*/2i (lambda is an element of T), designated as the generalized Cartesian decomposition of T. Using the generalized Cartesian decompositionwe obtain several lower and upper bounds for the numerical radius of bounded linear operators which refine the existing bounds. We prove that if T is a bounded linear operator on H, then w(T) >= 1/2 ||T + lambda + mu/2 T*||, for all lambda, mu is an element of T. This improves the existing bounds w(T) >= 1/2 ||T||, w(T) >= ||Re(T)||, w(T) >= ||Im(T)|| and so w(2)(T) >= 1/4 ||T*T + TT*||, where Re(T) and Im(T) denote the the real part and the imaginary part of T, respectively. Further, using a lower bound for the numerical radius of a bounded linear operator, we develop upper bounds for the numerical radius of the commutator of operators which generalize and improve on the existing ones.