Refinement of numerical radius inequalities of complex Hilbert space operators.

We develop upper and lower bounds for the numerical radius of 2 × 2 off-diagonal operator matrices, which generalize and improve on some existing ones. We also show that if A is a bounded linear operator on a complex Hilbert space, then for all r ≥ 1 , w 2 r (A) ≤ 1 4 ‖ | A | 2 r + | A ∗ | 2 r ‖ + 1 2 min ‖ ℜ (| A | r | A ∗ | r) ‖ , w r (A 2) where w(A), ‖ A ‖ and ℜ (A) , respectively, stand for the numerical radius, the operator norm and the real part of A. This (for r = 1 ) improves on some existing well-known numerical radius inequalities. [ABSTRACT FROM AUTHOR]