Combinatorial methods for the spectral p-norm of hypermatrices.

The spectral p -norm of r -matrices generalizes the spectral 2-norm of 2-matrices. In 1911 Schur gave an upper bound on the spectral 2-norm of 2-matrices, which was extended in 1934 by Hardy, Littlewood, and Pólya to r -matrices. Recently, Kolotilina, and independently the author, strengthened Schur's bound for 2-matrices. The main result of this paper extends the latter result to r -matrices, thereby improving the result of Hardy, Littlewood, and Pólya. The proof is based on combinatorial concepts like r-partite r-matrix and symmetrant of a matrix, which appear to be instrumental in the study of the spectral p -norm in general. Thus, another application shows that the spectral p -norm and the p -spectral radius of a symmetric nonnegative r -matrix are equal whenever p ≥ r . This result contributes to a classical area of analysis, initiated by Mazur and Orlicz back in 1930. Additionally, a number of bounds are given on the p -spectral radius and the spectral p -norm of r -matrices and r -graphs. [ABSTRACT FROM AUTHOR]