Nonlinear Theory

Abstract

The work of the preceding two chapters has dealt exclusively with finite sections of those classical inequalities which describe the bound of a linear operator on l ². Yet there remains the tantalizing inequality of Theorem 1.38

$$\begin{array}{*{20}{c}} {\left( a \right){{{\sum\limits_{{n = 1}}^{\infty } {\left( {\sum\limits_{{m = 1}}^{\infty } {K\left( {m,n} \right){{x}_{m}}} } \right)} }}^{p}} < {{k}^{p}}\sum {x_{m}^{p}} } \hfill \\ {\left( b \right)k = \int\limits_{0}^{\infty } {K\left( {x,1} \right){{x}^{{ - 1/p}}}dx} } \hfill \\ \end{array}$$

((4.1))

in which K(x, y) is homogeneous of degree -1 and the constant k is the best possible one. We have found that the case p = 2 of (4.1) is already essentially equivalent to the theory of sections of Toeplitz forms. We further found that when p = 2 the constant k of (4.1) (b) is just one superficial manifestation of the deeper fact that the spectral theory of the sections depends upon the function

$$k = \int\limits_{0}^{\infty } {K(x,1){{x}^{{ - s}}}dx}$$

((4.2))

for s = 1/2 + iξ. One may expect that the sections of (4.1) will depend upon (4.2) for s = 1/p + iξ, but this has not been proved. Indeed, for p ≠ 2 one has available none of the vast resources of linear algebra which are available in the self-dual situation, and recourse must be had to the method of hard analysis.