Spectrum-preserving elementary operators on \(B(X)\) (Q1273112)

Let \(X\) be an infinite-dimensional complex Banach space and \(B(X)\) the Banach algebra of all bounded linear operators on \(X\). For \(T\in B(X)\), \(\sigma(T)\), as usual, will denote the spectrum of \(T\). Let \(\Phi\) be a linear map from \(B(X)\) into itself. \(\Phi\) is spectrum-preserving if \(\sigma(\Phi(T))= \sigma(T)\) for all \(T\in B(X)\); \(\Phi\) is spectrum-compressing if \(\sigma(\Phi(T))\subseteq \sigma(T)\) for all \(T\in B(X)\). It is clear that if \(\Phi\) is unital (i.e., \(\Phi(I)= I\)), then \(\Phi\) is spectrum-preserving (spectrum compressing) if and only if \(\Phi\) preserves invertibility in both directions (preserves invertibility), i.e., \(\Phi(T)\) is invertible if and only if \(T\) is (\(\Phi(T)\) is invertible if \(T\) is). Spectrum-preserving linear maps have been studied by many authors. In fact, this is one of the so-called linear preserver problems. It is interesting to ask if one can give a characterization for the structure of spectrum-preserving linear maps which are not surjective. But this question seems very difficult to answer. An important class of linear maps on \(B(X)\) which contains many non-surjective maps is the class of elementary operators. Recall that \(\Phi\) is called an elementary operator if there exist operators \(A_1,\dots, A_n\), \(B_1,\dots, B_n\) in \(B(X)\) such that \[ \Phi(T)= \sum^n_{i=1} A_iTB_i\quad\text{for all }T\in B(X). \] The number \(\ell(\Phi)= \inf\{n; \Phi(.)= \sum^n_{i=1} A_i(.)B_i\}\) is called the length of \(\Phi\). The main result of this paper is the following theorem: Theorem. Let \(\Phi(.)= A_1(.)B_1+ A_2(.)B_2\) be an elementary operator of length 2 with \(I\in R(\Phi)\). Then the following statements are equivalent. (i) \(\Phi\) is spectrum-compressing; (ii) \(\Phi\) is spectrum-preserving; (iii) \(A= (A_1,A_2)\) is invertible with \(A^{-1}= \left(\begin{smallmatrix} B_1\\ B_2\end{smallmatrix}\right)\); (iv) There exists an invertible operator \(A\in B(X\oplus X,X)\) such that \(\Phi(T)= AT\oplus TA^{-1}\) for all \(T\).