W_A Contractions

Abstract

The problem of unitary ρ-dilation can be generalized by Langer [9, p.55] as follows: Let A be a positive linear operator on a Hilbert space H, 0 < mI ≤ A ≤ MI, and C_A = {T : QTⁿQ = P_HUⁿ|_H(n = 1,2,3,...) where Q = A^-1/2 and U is a unitary on some Hilbert space H₁ ⊃ H}. Then T ∈ C_A if and only if T satisfies the condition: A + 2Re z(I - A)T + |z|²T*(A - 2I)T ≥ 0. Using the above generalization, we have a block-matrix criterion for an element in C_A as follows: T ∈ C_A if and only if P(A,z,T,n) ≥ 0(n = 1,2,3,...) [Theorem 2.5]. We define the operator radii w_A(.) by w_A(T) = inf;{r>0 : T/r ∈ C_A}. Applying the block-matrix criterion, we give some fundamental properties for w_A(.) and extend some earlier results involving operator radii w_ρ(.)(ρ > 0) in Fong and Holbrook (1983), Haagerup and de la Harpe (1992), Holbrook (1968), Holbrook (1969) and Holbrook (1971) to the case of w_A(.). We have the equalities $w_\rho (T) = \inf \{ r > 0:\rho ^{ - 1} rQ(\rho ,1,r^{ - 1} T,n) \geqslant 0{\text{ for all }}n = 1,2,3,...\} (\rho > 0)$ and $w_\rho (T) = \inf \{ ||B||:w_\rho (B^{ - 1/2} TB^{ - 1/2} ) \leqslant 1,B > 0\} (0 < \rho \leqslant 2)$. Inequalities involving completely bounded linear maps on unital C*-algebras are also provided [Theorem 4.5].