Additive spectrum compressors (Q1772280)

Let \(X\) be a Banach space and denote with \(B(X)\) and \(F(X)\) the algebra of all bounded linear operators on \(X\) and the ideal of all of its finite rank elements, respectively. By a unital standard operator algebra on \(X\), the author means a closed unital subalgebra of \(B(X)\) which contains \(F(X)\). The aim of this paper is to describe the general form of surjective additive maps between unital standard operator algebras which compress the spectrum. The main result (Theorem 7) solves a somewhat more complicated problem and it reads as follows. Let \(X\), \(Y\) be complex Banach spaces and let \(\mathcal A\), \(\mathcal B\) be unital standard operator algebras on \(X\) and \(Y\), respectively. Suppose that \(\dim\mathcal A=\infty\). Let \(\Phi:\mathcal A\to\mathcal B\) be a surjective additive map which does not annihilate all rank-one idempotents. Suppose that for each \(S\in \mathcal A\) we have either \[ \text{Sp}_{\mathcal B}(\Phi(S))\subset \text{Sp}_{\mathcal A}(S) \] or \[ \text{Sp}_{\mathcal B}(\Phi(S))\subset\overline{\text{Sp}_{\mathcal A}(S)}, \] where \(\text{Sp}\) denotes spectrum and bar stands for complex conjugate. Then \(\Phi\) takes one of the following two standard forms: \[ \Phi(S)=ASA^{-1} \quad (S\in \mathcal A), \] \[ \Phi(S)=BS'B^{-1} \quad (S\in\mathcal A), \] where \(A:X\to Y\) and \(B:X^*\to Y\) are bounded (conjugate) linear bijections, respectively. In the latter case, \(X\) and \(Y\) are reflexive. The proof uses some important facts from the theory of analytic functions. An application of the result above to so-called local approximate (anti)multiplications is also presented.