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

Suppose H₁, H₂, , Hₙ are arbitrary complex Hilbert spaces, and A=A₈₉ is an n n operator matrix with A₈₉ B (Hⱼ, Hᵢ). We show that w (A) w (bmatrix a₈₉ bmatrix₈, ₉=₁ⁿ), where w () denotes the numerical radius and the entries a₈₉=cases w (A₈₈) & if i=j, (\|A₈₉\|+\|₀_₉₈\|) ²- (\|A₈₉\| \|A₉₈\|-w (A₉₈A₈₉) ) ^ & if ij. cases This bound improves w (A) w (bmatrix a'₈₉ bmatrix₈, ₉=₁ⁿ), where a'₈₉=w (A₈₈) if i=j and a'₈₉=\|A₈₉\| 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 (A₁B₁+A₂B₂) 1 2 (w (B₁A₁) +w (B₂A₂) ) + 1 2 (w (B₁A₁) -w (B₂A₂) ) ² + 3\|B₁A₂\|\|B₂A₁\| +, eqnarray* where =w (B₂A₁ B₁A₂), and r () denotes the spectral radius. We also achieve a bound for the roots of an algebraic equation.