Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1131.47035
Cui, Jianlian
Additive Drazin inverse preservers. (English)
[J] Linear Algebra Appl. 426, No. 2-3, 448-453 (2007). ISSN 0024-3795

Let $H$ be a real or complex Hilbert space and denote by $B(H)$ the algebra of all bounded linear operators acting on $H$. An element $T\in B(H)$ is called Drazin invertible if there exists an element $T^D\in B(H)$ and a positive integer $k$ such that $$ TT^D=T^DT, \quad T^DTT^D=T^D, \quad T^{k+1}T^D=T^k. $$ The operator $T^D$ is unique and called the Drazin inverse of $T$. The author characterizes the additive maps $\phi:B(H)\to B(K)$ ($H,K$ being infinite-dimensional real or complex Hilbert spaces) which preserve the Drazin inverse in the sense that $\phi(T^D)=\phi(T)^D$ holds for every Drazin invertible operator $T\in B(H)$. It is proved that if the range of $\phi$ contains every rank-one idempotent in $B(K)$ and $\phi$ does not annihilate all rank-one idempotents in $B(H)$, then $\phi$ is of one of the forms $$ \phi(T)=\xi ATA^{-1}, \quad A\in B(H), $$ $$ \phi(T)=\xi AT^{tr}A^{-1}, \quad A\in B(H), $$ where $\xi=\pm 1$ and $A:H\to K$ is a bounded linear or conjugate-linear bijection. The finite-dimensional case is also considered.
[Lajos Molnár (Debrecen)]

MSC 2000:: *47B49 Transformers (=operators on spaces of operators)
47A05 General theory of linear operators

Keywords: additive preservers; Drazin inverse of operators

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]