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On the polynomial numerical hull of a normal matrix. (English) Zbl 1207.15023
The polynomial numerical hull of order $k$ of an $n$-by-$n$ complex matrix $A$ is, by definition, the set $V^k(A)= \{z\in\bbfC:|p(z)|\le\Vert p(A)\Vert$ for all complex polynomials $p$ with degree at most $k\}$. It is known that $V^k(A)= \{z\in\bbfC: (z,\dots, z^k)\in W(A,\dots, A^k)^\wedge\}$, where $W(A,\dots, A^k)^\wedge$ denotes the convex hull of the joint numerical range $W(A,\dots, A^k)= \{(x^* Ax,\dots, x^*A^kx): x\in\bbfC^n$, $x^* x= 1\}$ of $A,\dots, A^k$. The main results of this paper address the open problem whether $V^k(A)$ equals the spectrum of $A$ for an $n$-by-$n$ normal matrix $A$ with at least $2k$ distinct eigenvalues and with $V^k(A)$ finite. Using the notion of the polynomial inverse image introduced by the authors and {\it A. Salemi} before [“Polynomial inverse images and polynomial numerical hulls of normal matrices”, Oper. Matrices, to appear], they can reduce this problem to the case of $n=2k$ and the spectrum lying on exactly one polynomial inverse image of order $k$. The paper ends with an algebraic proof of a key theorem in determining $V^2(A)$ for a normal $A$ obtained before by {\it C. Davis}, {\it C.-K. Li} and {\it A. Salemi} [Linear Algebra Appl. 428, No. 1, 137--153 (2008; Zbl 1130.15017), Theorem 3.2]. We remark that the characterizations of $V^k(A)$ and its boundary for bounded operators $A$ on a Hilbert space by {\it J. V. Burke} and {\it A. Greenbaum} [Linear Algebra Appl. 419, No. 1, 37--47 (2006; Zbl 1106.15018), Theorems 3.1 and 3.2] can be used to derive some of the results here. Moreover, the paper is full of obvious English grammatical errors. The concerned editor or the editorial office of the journal should play a more active role on having them corrected before publication.

##### MSC:
 15A60 Applications of functional analysis to matrix theory 15A18 Eigenvalues, singular values, and eigenvectors 47A12 Numerical range and numerical radius of linear operators 15B57 Hermitian, skew-Hermitian, and related matrices
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