Strengthening of spectral radius, numerical radius, and Berezin radius inequalities

Suppose H₁, H₂, …, Hₙ are arbitrary complex Hilbert spaces, and A=[Aᵢⱼ] is an n× n operator matrix with Aᵢⱼ∈ B(Hⱼ, Hᵢ). We show that w( A) ≤ w(bmatrix aᵢⱼ bmatrixi,j=1ⁿ ), where w(·) denotes the numerical radius and the entries aᵢⱼ=cases w(Aᵢᵢ) & ifi=j, √ ( \|Aᵢⱼ\|+\|Aⱼᵢ\| )²- (\|Aᵢⱼ\| \|Aⱼᵢ\|-w(AⱼᵢAᵢⱼ) )^ & ifi<j, 0 & ifi>j. cases This bound improves w( A) ≤ w(bmatrix a'ᵢⱼ bmatrixi,j=1ⁿ ), where a'ᵢⱼ=w(Aᵢᵢ) if $i=j$ and a'ᵢⱼ=\|Aᵢⱼ\| if i≠ j. We deduce an upper bound for the Kronecker products A⊗ B, where A∈ Mₙ(C) and B∈ B(H₁), which refines Holbrook's classical bound w(A⊗ B)≤ w(A)\|B\|, when all entries of A are non-negative. Further, we obtain the Berezin radius inequalities for n× n operator matrices where the entries are reproducing kernel Hilbert space operators. We provide an example, which illustrates these inequalities for some concrete operators on the Hardy--Hilbert space. Applying the numerical radius bounds, we show that if Aᵢ ∈ B(Hᵢ, H₁) and Bᵢ∈ B(H₁, Hᵢ) for $i=1,2,$ then {eqnarray*} r(A_1B_1+A_2B_2) ≤ 1 /2 (w(B_1A_1)+w(B_2A_2) ) + 1 /2 √ (w(B_1A_1)-w(B_2A_2))^2 + 3\|B_1A_2\|\|B_2A_1\| + η, {eqnarray*} where η=w(B₂A₁ B₁A₂), and r(·) denotes the spectral radius. We also achieve a bound for the roots of an algebraic equation.