Linear maps compressing the polynomial numerical hulls of matrices
Let \(n\geq 2\) be an integer and let \(\mathcal{M}_{n}\) be the algebra of all complex \(n\times n\) matrices. By \(T^{t}\in \mathcal{M}_{n}\) we denote the transpose and by \(T^{\ast }\in \mathcal{M}_{n}\) the conjugate transpose of \(T\in \mathcal{M}_{n}\). Let \(\sigma (T)\) denote the spectrum of \(T\in \mathcal{M}_{n}\). For \(T\in \mathcal{M}_{n}\), we define its numerical range as \[ w(T)=\left\{ x^{\ast }Tx:x\in \mathbb{C}^{n},\left\Vert x\right\Vert =1\right\} . \] Let \(\mathcal{P}_{k}\) be the set of all polynomials with complex coefficients of degree at most \(k\) and let \(\left\Vert .\right\Vert\) denote the operator norm in \(\mathcal{M}_{n}\). The polynomial numerical hull of order \(k\geq 1\) of \(T\in \mathcal{M}_{n}\) is defined as \[ V^{k}(T)=\left\{ \lambda \in \mathbb{C}:\left\vert p(\lambda )\right\vert \leq \left\Vert p(T)\right\Vert \text{ for every }p\in \mathcal{P}_{k}\right\} . \] Note that the concept of polynomial numerical hull extends the notion of numerical range of \(T\in \mathcal{M}_{n}\), namely \(V^{1}(T)=w(T)\). In [Linear Algebra Appl. 575, 27--34 (2019; Zbl 1416.15021)], \textit{Gh. Aghamollaei} et al. characterized linear maps on \(\mathcal{M}_{n}\) which preserve the polynomial numerical hull of matrices of some fixed order \(k\). It turns out that for a linear map \(\Phi : \mathcal{M}_{n}\rightarrow \mathcal{M}_{n}\) and an integer \(k\) with \(1\leq k<n\) we have \[ V^{k}\left( \Phi (T)\right) =V^{k}(T)\quad \text{for every }T\in \mathcal{M}_{n}, \] if and only if there exists a unitary matrix \(U\in \mathcal{M}_{n}\) such that either \[ \Phi (T)=UTU^{\ast }\quad \text{for every }T\in \mathcal{M}_{n} \tag{1} \] or \[ \Phi (T)=UT^{t}U^{\ast }\quad \text{for every }T\in \mathcal{M}_{n}. \tag{2} \] In the present paper, the author studies linear maps \(\Phi\) on \(\mathcal{M}_{n}\) which compress the polynomial numerical hull of order \(k\), i.e., \[ V^{k}\left( \Phi (T)\right) \subseteq V^{k}(T)\quad \text{for every }T\in \mathcal{M}_{n}. \tag{3} \] He proves that when \(1<k<n\) and \(n\geq 3\), a linear map \(\Phi : \mathcal{M}_{n}\rightarrow \mathcal{M}_{n}\) satisfies (3) if and only if there exists a unitary matrix \(U\in \mathcal{M}_{n}\) such that \(\Phi\) is either of the form (1) or (2). The same conclusion holds if we replace (3) with \(V^{k}\left( \Phi (T)\right) \supseteq V^{k}(T)\) for every \(T\in \mathcal{M}_{n}\). An example is presented which shows that if \(k=1\) we need to impose additional assumptions on the numerical range in order to have the compressing linear map \(\Phi\) in the form (1) or (2). The following result is then proved. Theorem. Suppose \(n\geq 2\) and let \(\Phi : \mathcal{M}_{n}\rightarrow \mathcal{M}_{n}\) be a linear map. Then \[ \sigma (T)\subseteq w(\Phi (T))\subseteq w(T)\quad \text{for every }T\in \mathcal{M}_{n} \tag{4} \] if and only if there exists a unitary matrix \(U\in \mathcal{M}_{n}\) such that \(\Phi\) is either of the form (1) or (2). The same conclusion holds when one replaces (4) with \[ \sigma (\Phi (T))\subseteq w(T)\subseteq w(\Phi (T))\quad \text{for every }T\in \mathcal{M}_{n}. \] The paper is concluded with some further remarks concerning linear maps \(\Phi\) on \(\mathcal{M}_{n}\) such that \(w(\Phi (T))\subseteq w(T)\) for each \(T\in \mathcal{M}_{n}\).
scientific article; zbMATH DE number 7462787
scientific article