ACTA issues

Linear maps that preserve the nilpotent operators

Peter Ċ emrl

Acta Sci. Math. (Szeged) 61:1-4(1995), 523-534

Abstract. Let $X$ be an infinite-dimensional Hilbert space. We prove that a surjective linear mapping $\phi\colon {\cal B}(X)\longrightarrow{\cal B}(X)$ that preserves the nilpotent operators in both directions is either of the form $\phi(T)=cATA^{-1}$ or of the form $\phi(T)=cAT^{tr}A^{-1}$, where $A$ is a bounded bijective linear operator on $X$, $c$ is a nonzero complex constant, and $T^{tr}$ denotes the transpose of $T$ relative to a fixed but arbitrary orthonormal basis. The description of all surjective linear mappings preserving nilpotent operators in both directions is slightly more complicated in the case that $X$ is an arbitrary Banach space.

AMS Subject Classification (1991): 47B49

Received April 28, 1994, and in revised form February 15, 1995. (Registered under 5702/2009.)