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

Let A=[Aᵢⱼ] be an n× n operator matrix where each Aᵢⱼ 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(Ã), where Ã=[aᵢⱼ] is an n× n complex matrix with {eqnarray*} aᵢⱼ={cases} w(Aᵢᵢ) if i=j\\ {0≤ t ≤ 1}{min} \| |Aᵢⱼ|²ᵗ + |Aⱼᵢ^*|²ᵗ \|1/2 \| |Aᵢⱼ^*|²⁽¹⁻ᵗ⁾+ |Aⱼᵢ|²⁽¹⁻ᵗ⁾ \|1/2 if i< j 0 if i> j. {cases} {eqnarray*} This bound refines the well known bound w(A) ≤ w(Â), where Â=[âᵢⱼ] is an n× n matrix with âᵢⱼ= w(Aᵢᵢ) if $i=j$ and âᵢⱼ= \|Aᵢⱼ\| 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.