Sharper bounds for the numerical radius of n n operator matrices II

Let A=A₈₉ be an n n operator matrix where each A₈₉ is a bounded linear operator on a complex Hilbert space H. With other numerical radius bounds via contraction operators, we show that w (A) w (A), where A=a₈₉ is an n n complex matrix with eqnarray* a₈₉=cases w (A₈₈) if i=j\\ 0 t 1 \| |A₈₉|^2t + |A₉₈^*|^2t \|^1/2 \| |A₈₉^*|^2 (1-t) + |A₉₈|^2 (1-t) \|^1/2 if i j. cases eqnarray* This bound refines the well known bound w (A) w (A), where A=a₈₉ is an n n matrix with a₈₉= w (A₈₈) if i=j and a₈₉= \|A₈₉\| if i j Linear Algebra Appl. 468 (2015), 18--26. We deduce that if A, B are bounded linear operators on H, then eqnarray* w (bmatrix 0&A\\ B&0 bmatrix) 12 \| |A|^2t + |B^*|^2t \|^1/2 \| |A^*|^2 (1-t) + |B|^2 (1-t) \|^1/2 for all t 0, 1. eqnarray* Further by applying the numerical radius bounds of operator matrices, we deduce some numerical radius bounds for a single operator, the product of two operators, the commutator of operators. We show that if A is a bounded linear operator on H, then w (A) 12 \|A\|ᵗ \| |A|^1-t+|A^*|^1-t \| for all t 0, 1, which refines as well as generalizes the existing ones.