The Generalized Inverse Of The Operator And The Quantum Effects

Opeator theory is one of the most important fields in functional analysis. In recentyears, Generalized Inverse and Effect Algebra have been live topics in operator theory. Theresearch of these subjects have related to pure and applied mathematics such as algebra,geometry, perturbation theory, matrix analysis, approximation theory, optimality principleand quantum physics etc. It is from the study in this field that we have a clear impressionon the internal relations and the constructions among operators. In this essey, we will dealwith the generalized Bott-Duffin inverses, genralized inverses of lower triangular operatorson a infinite dimensional Hilbert space and the sequential product of quantum effect on afinite dimensional Hilbert space. This article is divided into three chapters.In Chapter 1, using the way of space decomposition and the technique of blockoperator matrices, we have given the matrix representations of the generalized Bott-Duffininverses of bounded linear operator A on a Hilbert space with respect to a closed subspaceM, and we extend some properties of generalized Bott-Duffin inverses of operator A withrespect to two closed complementary subspaces M and M of finite dimensional Hilbertspace V., give new proofs. These block operators matrix methods not only make proofclearer, but also give us more information about geometrical structures of generalizedBott-Duffin inverses of operators.In Chapter 2,we pay more attentions to the lower triangular operator matrix of 2×2.Let H and K. be complex separable Hilbert spaces, B(H), B(K) and B(H,K) denote theset of all bounded linear operators on H, Kand from H into K, respectively. For given(?), (?), suppose (?) let Mc denote the 2×2 lower triangularoperatormatrix: (?)in this chapter,we mainly answer the following two questions:(1)Under what conditions, the genralized inverses of the lower triangular operatorsMe exists?(2)If the generalized inverses of Me exists, give the matix representations of it?In Chapter 3, we applied the idea of block operator matrix to the proof of the followingtwo coiiclussins and give a more elementary and acceptable proof. Let (?) denote theeffect algebra on a Hilbert space H and (?). Suppose that dim H(?). then(1)(?)(2)(?) Thismethod we used in the proof can be understand and accepted. trix quantum effect, sequential product...