Study On Some Problems Of 2-local Lie Derivations Of Triangular Algebras And Product Of Orthogonal Projection

In this paper,The issues of 2-local Lie derivations of triangular algebras and orthogonal projections are studied.First of all,Every additive 2-local Lie derivation of triangular algebras is the sum of an additive derivation and a additive map is proved.Then,There is topics on the norm estimation,products and invertibility of two orthogonal projections.Projections and operator spectra are heated topics in operator theory and also have important value in both theory and application.Using the technique of operator blocks,the internal relations and constructions among operators can be found.This article is divided into four parts.Firstly,Literature review,preliminary knowledge and main results are given.Next,Every additive 2-local Lie derivation of triangular algebras is the sum of an additive derivation and a additive map into its center sending commutators to zero is proved.Then,Based on space decomposition、the block operator technique and operator spectral theory,‖PX-XQ‖= max{‖X12‖,‖X21‖} and ‖P-Q‖ ≤1 are proved.Particularly,some sufficient conditions for ‖P-Q‖=1 and ‖P-Q‖<1 is discussed.At the same time,an equivalent condition for invertibility of P-Q is given.Finally,there are the set x and y of products of orthogonal projections PQ and PQP,Give different optimal proof methods of two orthogonal projective products,using operator blocking techniques to avoid unbounded operators.And the necessary and sufficient conditions for the integral solutions of two orthogonal projections are also given.Using the operator blocking technique to clearly illustrate all possible relationships among ‖P-Q‖、‖Q(I-P)‖ and ‖P(I=Q)‖,and illustrates the relationship between R(S)and ‖P-Q‖=1、‖P-Q‖<1,where R(S)∈ y.