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Linear Combinations, Path Connectivity of Idempotents and Drazin Inverse in B(H)

Author: ChenYanNi
Tutor: DuHongKe
School: Shaanxi Normal University
Course: Basic mathematics
Keywords: Orthogonal projection Idempotent operator Kovarik formula Drazin inverse
CLC: O177
Type: Master's thesis
Year: 2006
Downloads: 72
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Abstract


Operator theory is more active the Drazin inverse operator and operator of an extremely important area of ??research, idempotent functional analysis operators in recent years of research topics. Their research involves many branches of basic and Applied Mathematics, such as algebra, geometry theory, operator perturbation theory, matrix theory, approximation theory, optimization theory and quantum physics, the operator can structure their the intrinsic relationship becomes clearer On the topic of the operator, but also makes a more solid theoretical foundation. This article covers a linear combination of the Hilbert space of two idempotent operators, the Drazin inverse path connection as well as operators in the Hilbert space three aspects. The paper is divided into three chapters, as follows: Chapter geometric decomposition theory based on space and count sub-block matrix techniques, given the power of the Hilbert space operator orthogonal projection operator, and use it as tools, system study in the power of the infinite-dimensional Hilbert space operator linear combination of nature, depicts the B (H) two idempotent operator P and Q are linear combinations of λ 1 P λ 2 Q to maintain the necessary and sufficient condition for idempotency which λ 1 λ 2 is nonzero complex, and to disseminate the literature [ 1] JKBaksalary OMBaksalary conclusion. It is worth noting that we come through rigorous reasoning, [1] in the theorem P 1 P 2 ≠ P 2 P < sub> 1 is not necessary. The second chapter focuses on infinite dimensional Hilbert space, the two homotopy idempotent operator connectivity issues. Kovarik formula proposed in 1977 by the ZVKovarik Because of this problem, has a pivotal role. In section II of this chapter, we discuss in depth the nature of the Kovarik formula for the generalized the Kovarik formula for the characteristic. Subsequently, as a tool, with the operator matrix partitioning techniques, given infinite dimensional Hilbert space homotopy idempotent operator (?) (Q) ≤ 2 when the sufficient said necessary conditions (?) (P, Q) connected in all idempotent operator P satisfy both connected from Q to P, holding power of the minimum number of line segments from P to Q, then [26] the in J.Giol given conditions to be more little weak compared to the conditions, we do concrete proof. Drazin inverse of the third chapter dedicated to the study defined operator in a Hilbert space. The use operator index theory and space decomposition theory, and by means of [35] said Drazin inverse of bounded linear operators in the Hilbert space, we prove that the two operators in infinite-dimensional Hilbert space P and Q PQ = 0 under the conditions, PQ is Drazin invertible and its corresponding expression of the Drazin inverse is given, so that the results are extended to the infinite-dimensional Hilbert space [40].

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