tailieunhanh - Báo cáo toán học: "Positive diagonal and triangular operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: nghiên cứu tích cực chéo và hình tam giác. | Copyright by INCREST 1980 J. OPERATOR THEORY 3 1980 165-178 POSITIVE DIAGONAL AND TRIANGULAR OPERATORS A. R. SCHEP INTRODUCTION In this paper we study two classes of order bounded operators on a Dedekind complete Riesz space. In Section 1 we consider order bounded operators with a strong local property the so called orthomorphisms which have been studied by several authors see . 3 11 22 and 23 . For a Dedekind complete Riesz space L the collection of all orthomorphisms is equal to I dd the band generated by the identity operator in the Riesz space of all order bounded operators on L. Hence every positive linear operator T L L has a unique decomposition T T with 0 T e I dd and 0 Tae I d. One can now consider Tỵ as the diagonal component of T and some of the results in Section 1 have been motivated by this point of view. In Section 2 we prove a continuity theorem for the spectral radius of a certain class of positive operators on a Banach lattice. In Section 3 we study a class of operators called triangular here which generalize the classical Volterra integral operators. An operator is called triangular if it has a maximal chain of invariant bands. An important result is then that every positive order continuous triangular compact operator with diagonal component zero is quasinilpotent. We also study the case that the diagonal component is not zero and prove that in that case the spectrum of the triangular operator is equal to the spectrum of its diagonal component. The author wishes to express his gratefulness to the referee for supplying shorter proofs for Lemma and Theorem . 1. ORTHOMORPHISMS Let L be an Archimedean Riesz space for terminology not explained here see 14 and 19 . A positive linear operator T from L into L is called a positive orthomorphism if 0 u lie L and u A V 0 implies Tu A V 0. A linear map from L into Lis now called an orthomorphism if it is the difference of two positive orthomorphisms. The set of orthomorphisms from L into L shall