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

Abstract

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.