# WA Contractions

WA Contractions 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]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# WA Contractions

, Volume 2 (4) – Oct 14, 2004
10 pages

/lp/springer_journal/wa-contractions-rbThoiqBsz
Publisher
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1009712101922
Publisher site
See Article on Publisher Site

### 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].

### Journal

PositivitySpringer Journals

Published: Oct 14, 2004

### References

• Multiplicative properties of the numerical radius in operator theory
Holbrook, J. A. R.

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