Some new results for Hua-type operator matrices

Let Ai(i=1,2,…,n) be strict contractions on a Hilbert space HH. The n×nn×n operator matrix Hn(A1,A2,⋯,An)=((I−Aj⁎Ai)−1)i,j=1n is called a Hua-type operator matrix. In this note, we mainly investigate some results which are related to the Hua-type operator matrix. We firstly give some equivalent conditions for the positivity of n×nn×n operator matrices (I−Aj⁎Ai)i,j=1n. Then the equation min{‖H2(A1,A2)‖:‖A1‖<1,‖A2‖<1}=2min{‖H2(A1,A2)‖:‖A1‖<1,‖A2‖<1}=2 is shown. In particular, some equivalent conditions for strict contractions A1A1 and A2A2 such that ‖H2(A1,A2)‖=2‖H2(A1,A2)‖=2 are obtained.