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On the norm of normal matrices (Research on preserver problems on Banach algebras and related topics)

BOUTHAT, Ludovick MASHREGHI, Javad MORNEAU-GUÉRIN, Frédéric 京都大学

2023.07

概要

In this article, we present some recent results related to the calculation of the induced p-norm of n × n circulant matrices A(n, a, b) with diagonal entries equal to a ∈ ℝ and off-diagonal entries equal to b ∈ ℝ. For circulant matrices with nonnegative entries, an explicit formula for the induced p-norm (1 ≤ p ≤ ∞) is given, whereas for A(n, −a, b), a > 0 the situation is no longer so simple and calls for a more subtle analysis. As a matter of fact, while the 2-norm of A(n, −a, b) is precisely determined, the exact value of the induced p-norm for 1 < p < ∞, p ≠ 2, still remains elusive. Nevertheless, we provide a lower bound as well as two different categories of upper bounds. As an indication of not being far from the exact values, our estimates coincide at both ends points (i.e., p = 1 and p = ∞) as well as at p = 2 with the precise values. As an abstract approach, we also introduce the ∗-algebra generated by a normal matrix A accompanied by an axis-oriented norm, and obtain some estimations of the norm of elements of the ∗-algebra. We then exhibit the connection between the new generalized estimates and the previously obtained estimates in the special case where A is a circulant matrix. Finally, using an optimization-oriented approach, we provide insight on the nature of the maximizing vectors for ∥Ax∥p/∥x∥p . This leads us to formulate a conjecture that, if proven valid, would make it possible to derive an exact formula for the induced p-norm of A(n, a, b) whenever a = 1−n/n and b = 1/n.

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