Sharp linear operators (Q1071239)

Let E be a Banach space, partially ordered by a wedge K. The linear operator A is called \(\phi\)-sharp \((0\leq \phi <\pi /2)\) if AK\(\subset K\) and \(f(A^ 2x)\geq | A^*f|_{E^*}| Ax|_ E\cos \phi\) for every x in K and f in \(K^*\), further \(f(A^ 2x)>0\) for some pair (x,f). The basic properties of a \(\phi\)-sharp operator A are listed in Theorem 1: A has an eigenvector e in K which corresponds to the simple eigenvalue \(r(A)>0\), the spectral radius of A. To every eigenvector in K that is not a multiple of e there corresponds the eigenvalue 0. For \(z\in \sigma (A)\), \(z\neq r(A)\) we have \(| z| \leq r(A)tg\phi /2.\) Theorems 2 and 3 give sufficient conditions that an operator A in \(R^ N\) determined by a matrix with nonnegative elements and an integral operator with continuous nonnegative kernel, respectively, be \(\phi\)- sharp for a certain angle \(\phi\). Theorem 4 states that the dual of a \(\phi\)-sharp operator is also \(\phi\)- sharp, and Theorem 5 establishes the connection between the notions of a \(\phi\)-sharp and of a focusing operator. Theorem 6 gives, loosely speaking, an estimation of the approximation of the (normalized) eigenvector \(e_ 0\) by a vector z satisfying certain conditions. No proofs are given.