2025/03/04 by Bhunia, Pintu · 1 citation
#15A60 #26C10 #47A10 #47A12 #47A30 #47B15 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2503.02615
Suppose H1, H2, …, Hn are arbitrary complex Hilbert spaces, and \bf A=[Aij] is an n× n operator matrix with Aij∈ B(Hj, Hi). We show that w(\bf A) ≤ w(\beginbmatrix aij \endbmatrixi,j=1n ), where w(⋅) denotes the numerical radius and the entries aij=\begincases w(Aii) amp; if i=j, √ ( ‖Aij‖+‖Aji‖ )2- (‖Aij‖ ‖Aji‖-w(AjiAij) ) amp; if ij. \endcases This bound improves w(\bf A) ≤ w(\beginbmatrix a'ij \endbmatrixi,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 r(A1B1+A2B2) ≤ ( 1 )/(2 ) (w(B1A1)+w(B2A2) ) + ( 1 )/(2 ) √( (w(B1A1)-w(B2A2))2 + 3‖B1A2‖‖B2A1‖ + η), where η=w(B2A1 B1A2), and r(⋅) denotes the spectral radius. We also achieve a bound for the roots of an algebraic equation.