Numerical radius inequalities of operator matrices

Suppose [A(ij)] is an nxn operator matrix, where each A(ij) is a bounded linear operator on a complex Hilbert space H. Among other inequalities, it is shown that w([A(ij)]) <= w([a(ij)]), where [a(ij)] is an nxn matrix with a(ij) = {w(A(ii)) if i=j, min(0 <= t <= 1) & Vert;|A(ij)|(2t)+|A(ji)& lowast;|2t & Vert;(1/2)& Vert;|A(ij)& lowast;|(2(1-t))+|A(ji)|2(1-t)& Vert;(1/2) if i>j, if i>j. This numerical radius bound refines a well known bound by Abu-Omar and Kittaneh [Linear Algebra Appl. 468 (2015), 18-26]. We use these estimates to derive several numerical radius inequalities and equalities for 2x2 operator matrices. Applying these inequalities, we also deduce several numerical radius bounds for a bounded linear operator, the product of two operators and the commutator of operators. In particular, it is shown that w(A) <= min(0 <= t <= )1 (1/2 & Vert;A & Vert;(t)& Vert;|A|(1-t)+|A & lowast;|(1-t)& Vert;), where A is a bounded linear operator on H. This bound refines as well as generalizes the well known bounds.