Abstract:

Let △ be the unit disk,and H₁(△) be the follwing set
H₁ (△)={φ(z):φ(z) is analytic in △,|φ(z)|<1,
and there exists a θ∈[0,2π]such that|φ(e^iθ<)|=1} Then all the proper contraction on H,we have
φ∈H₁^sup(△) b²(φ)=2,φ∈H₁^sup(△)b²(φ)=1 where b²(φ)=sup_A(||φ'(A)||(1-||A||²)+||φ(A)||²)
b²(φ)=sup_A(||φ'(A)||(1-||A||²)+inf_x∈H^||x||=1||φ(A)x||²)On the other hand,if φ(z)=Σ_k=0ⁿ a_kz_k,|φ(z)|<1, then
||φ'(A)||+||φ'(A)|| ≤ 2n where φ(z)=zⁿφ(z^-1), and 2 is the best constant for all the choices of n,φ(z) and A.

Key words: Proper contraction, operator polynomial unitary dilatation