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 CA = {T : QTnQ = PHUn|H(n = 1,2,3,...) where Q = A-1/2 and U is a unitary on some Hilbert space H1 ⊃ H}. Then T ∈ CA 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 CA as follows: T ∈ CA if and only if P(A,z,T,n) ≥ 0(n = 1,2,3,...) [Theorem 2.5]. We define the operator radii wA(.) by wA(T) = inf;{r>0 : T/r ∈ CA}. Applying the block-matrix criterion, we give some fundamental properties for wA(.) and extend some earlier results involving operator radii wρ(.)(ρ > 0) in Fong and Holbrook (1983), Haagerup and de la Harpe (1992), Holbrook (1968), Holbrook (1969) and Holbrook (1971) to the case of wA(.). 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].
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Suen, CY. WA Contractions. Positivity 2, 301–310 (1998). https://doi.org/10.1023/A:1009712101922
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DOI: https://doi.org/10.1023/A:1009712101922