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

Loading next page...
 
/lp/springer_journal/wa-contractions-rbThoiqBsz
Publisher
Kluwer Academic Publishers
Copyright
Copyright © 1998 by Kluwer Academic Publishers
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

References

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

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

Monthly Plan

  • Read unlimited articles
  • Personalized recommendations
  • No expiration
  • Print 20 pages per month
  • 20% off on PDF purchases
  • Organize your research
  • Get updates on your journals and topic searches

$49/month

Start Free Trial

14-day Free Trial

Best Deal — 39% off

Annual Plan

  • All the features of the Professional Plan, but for 39% off!
  • Billed annually
  • No expiration
  • For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.

$588

$360/year

billed annually
Start Free Trial

14-day Free Trial