# Daugavet type inequalities for operators on L p–spaces

Daugavet type inequalities for operators on L p–spaces Let T be a regular operator from L p → L p. Then $$T \bot I{\text{ implies that }}\left\| {I \pm T} \right\|_r \geqslant (1 + \left\| T \right\|_r^p )^{\frac{1}{p}}$$ , where ∥T∥r denotes the regular norm of T, i.e., ∥T∥r=∥ |T| ∥ where |T| denotes the modulus operator of a regular operator T. For p=1 every bounded linear operator is regular and ∥T∥=∥T∥r, so that the above inequality generalizes the Daugavet equation for operators on L 1–spaces. The main result of this paper (Theorem 9) is a converse of the above result. Let T be a regular linear operator on L p and denote by T A the operator T○χA. Then $$\left\| {I_A \pm T_A } \right\|_r \geqslant (1 + \left\| {T_A } \right\|_r^p )^{\frac{1}{p}}$$ for all A with μ(A)>0 if and only if $$T \bot I$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Daugavet type inequalities for operators on L p–spaces

Positivity, Volume 7 (2) – Oct 17, 2004
9 pages

/lp/springer_journal/daugavet-type-inequalities-for-operators-on-l-p-spaces-pL6KXIewkf
Publisher
Springer Journals
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:1025831215425
Publisher site
See Article on Publisher Site

### Abstract

Let T be a regular operator from L p → L p. Then $$T \bot I{\text{ implies that }}\left\| {I \pm T} \right\|_r \geqslant (1 + \left\| T \right\|_r^p )^{\frac{1}{p}}$$ , where ∥T∥r denotes the regular norm of T, i.e., ∥T∥r=∥ |T| ∥ where |T| denotes the modulus operator of a regular operator T. For p=1 every bounded linear operator is regular and ∥T∥=∥T∥r, so that the above inequality generalizes the Daugavet equation for operators on L 1–spaces. The main result of this paper (Theorem 9) is a converse of the above result. Let T be a regular linear operator on L p and denote by T A the operator T○χA. Then $$\left\| {I_A \pm T_A } \right\|_r \geqslant (1 + \left\| {T_A } \right\|_r^p )^{\frac{1}{p}}$$ for all A with μ(A)>0 if and only if $$T \bot I$$ .

### Journal

PositivitySpringer Journals

Published: Oct 17, 2004

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