For an n-by-n matrix A, let f A be its "field of values generating function" defined as f A : x → x * Ax. We consider two natural versions of the continuity, which we call strong and weak, of f −1 A (which is of course multivalued) on the field of values F (A). The strong continuity holds, in particular, on the interior of F (A), and at such points z ∈ ∂F (A) which are either corner points, belong to the relative interior of flat portions of ∂F (A), or whose preimage under f A is contained in a one-dimensional set. Consequently, f −1 A is continuous in this sense on the whole F (A) for all normal, 2-by-2, and unitarily irreducible 3-by-3 matrices. Nevertheless, we show by example that the strong continuity of f −1 A fails at certain points of ∂F (A) for some (unitarily reducible) 3-by-3 and (unitarily irreducible) 4-by-4 matrices. The weak continuity, in its turn, fails for some unitarily reducible 4-by-4 and untiarily irreducible 6-by-6 matrices.
Abstract. Let A ∈ Mn(C). We consider the mapping f A (x) = x * Ax, defined on the unit sphere in C n . The map has a multi-valued inverse f A are considered in terms of the structure of the set of pre-images for points in the numerical range. It is shown that there may be only finitely many failures of continuity of f −1 A , and conditions for where these failure occur are given. Additionally, we give a necessary and sufficient condition for weak inverse continuity to hold for n = 4 and a sufficient condition for n > 4.
Abstract. This paper considers matrices A ∈ M n (C) whose numerical range contains boundary points generated by multiple linearly independent vectors. Sharp bounds for the maximum number of such boundary points (excluding flat portions) are given for unitarily irreducible matrices of dimension 5 . An example is provided to show that there may be infinitely many for n = 6 . For matrices unitarily similar to tridiagonal, however, a finite upper bound is found for all n . A somewhat unexpected byproduct of this is an explicit example of A ∈ M 5 (C) which is not tridiagonalizable via a unitary similarity.Mathematics subject classification (2010): Primary 15A60, 47A12; Secondary 54C08.
We study the dynamics of fixed point free mappings on the interior of a normal, closed cone in a Banach space that are nonexpansive with respect to Hilbert's metric or Thompson's metric. We establish several Denjoy-Wolff type theorems that confirm conjectures by Karlsson and Nussbaum for an important class of nonexpansive mappings. We also extend and put into a broader perspective results by Gaubert and Vigeral concerning the linear escape rate of such nonexpansive mappings.
We prove that there exists an exponent beyond which all continuous conventional powers of n-by-n doubly nonnegative matrices are doubly nonnegative. We show that this critical exponent cannot be less than n − 2 and we conjecture that it is always n − 2 (as it is with Hadamard powering). We prove this conjecture when n < 6 and in certain other special cases. We establish a quadratic bound for the critical exponent in general.2000 Mathematics Subject Classification. Primary 15Axx.
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