𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Rank one operators and norm of elementary operators

✍ Scribed by Ameur Seddik

Publisher
Elsevier Science
Year
2007
Tongue
English
Weight
141 KB
Volume
424
Category
Article
ISSN
0024-3795

No coin nor oath required. For personal study only.

✦ Synopsis

Let A be a standard operator algebra acting on a (real or complex) normed space E. For two n-tuples A = (A 1 , . . . , A n ) and B = (B 1 , . . . , B n ) of elements in A, we define the elementary operator R A,B on A by the relation R A,B (X) = n i=1 A i XB i for all X in A. For a single operator A ∈ A, we define the two particular elementary operators L A and R A on A by L A (X) = AX and R A (X) = XA, for every X in A. We denote by d(R A,B ) the supremum of the norm of R A,B (X) over all unit rank one operators on E. In this note, we shall characterize: (i) the supremun d(R A,B ), (ii) the relation

Moreover, we shall show the lower estimate d(L A -R B ) max{sup λ∈V (B) A -λI , sup λ∈V (A) B -λI } (where V (X) is the algebraic numerical range of X in A).

πŸ“œ SIMILAR VOLUMES

Norms and CB norms of Jordan elementary
Norms and CB norms of Jordan elementary operators
✍ Richard M. Timoney πŸ“‚ Article πŸ“… 2003 πŸ› Elsevier Science 🌐 French βš– 136 KB

We establish lower bounds for norms and CB-norms of elementary operators on B(H ). Our main result concerns the operator T a,b x = axb + bxa and we show T a,b a b , proving a conjecture of M. Mathieu. We also establish some other results and formulae for T a,b cb and T a,b for special cases.

Resonances and residue operators for sym
Resonances and residue operators for symmetric spaces of rank one
✍ J. Hilgert; A. Pasquale πŸ“‚ Article πŸ“… 2009 πŸ› Elsevier Science 🌐 English βš– 195 KB

Let X = G/K be a rank-one Riemannian symmetric space of the noncompact type and letbe the Laplace-Beltrami operator on X. We show that the resolvent operator R(z) of can be meromorphically continued across the spectrum and explicitly determine the poles, i.e. the resonances. Further we describe the

Additive mappings that preserve rank one
Additive mappings that preserve rank one nilpotent operators
✍ Wu Jing; Pengtong Li; Shijie Lu πŸ“‚ Article πŸ“… 2003 πŸ› Elsevier Science 🌐 English βš– 113 KB

Let B(X) be the algebra of bounded operators on a complex infinite dimensional Banach space X, and F (X) be the subalgebra of all finite rank operators in B(X). A characterization of additive mappings on F (X) which preserve rank one nilpotent operators in both directions is given. As applications o