Equivalence constants for certain matrix norms II

We give an almost complete solution of a problem posed by Klaus and Li A.-L. Klaus, C.-K. Li, Isometries for the vector (p, q) norm and the induced (p, q) norm, Linear and Multilinear Algebra 38 (1995) 315–332]. Klaus and Li’s problem, which arose during their investigations of isometries, was to relate the Frobenius (or Hilbert–Schmidt) norm of a matrix to various operator norms of that matrix. Our methods are based on earlier work of Feng B.Q. Feng, Equivalence constants for certain matrix norms, Linear Algebra Appl. 374 (2003) 247–253] and Tonge A. Tonge, Equivalence constants for matrix norms: a problem of Goldberg, Linear Algebra Appl. 306 (2000) 1–13], but introduce as a new ingredient some techniques developed by Hardy and Littlewood G.H. Hardy, J.E. Littlewood, Bilinear forms bounded in space p, q], Quart. J. Math. (Oxford) 5 (1934) 241–254].