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Schauder basis and operator theory

Author: ZhangJing
Tutor: CaoYang
School: Jilin University
Course: Basic mathematics
Keywords: Schauder basis Riesz basis Schauder operator Schauder matrix
CLC: O177.1
Type: Master's thesis
Year: 2011
Downloads: 18
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Abstract


On Banach spaces and Hilbert spaces in the learning process, the base is a very useful and important tool. Schauder base since 1927 after being J.schauder raised, appeared on the Schauder basis of numerous articles and books. Nowadays this area Research has formed a specialized field, but the operator theory and Schauder basis of binding studies are very rare. Olevskii article \The combination of a masterpiece. English translation of the article made a serious low-level errors, but also to professionals who review it does not move the word actually repeated the mistake, has also appeared in the literature review. Below is Comments Entry: MR0318848 (47 # 7394) Olevskii, AM Operators that produce conditional basis in a Hilbert space. (Russian) Mat. Zametki 12 (1972), 7384. (Reviewer: M. Cirnu), 46C10The author obtains a spectral characterization for the linear operators that transform every complete orthonormal system into a conditional basis in a Hilbert space. Section II of this paper mainly focus on the existence of Schauder basis, Schauder base sequence of existence, unconditional basis and unconditional base sequence introduces some basic definitions, etc. and results of the third, fourth, five are the main contents of this paper. around Olevskii discussed above in this article and those dramatic errors, trying to do two things: 1. pointed out the error, and gives the corresponding proof and detailed characterization; 2. collected relevant results, introduction Olevskii core theorem Definition 1.1 A bounded linear operator T ∈ B (H) is called a Schauder operator, if there is an ONB (?) that T ∞ × ∞ on its column vectors of the matrix form a Schauder based sequences. Definition 1.2 A ∞ × ∞ matrix called a Schauder matrix, if it is a column vector {ξk} k = 0 ∞ Schauder group form a sequence. Theorem 1.1. compact operator K is Schuader operator if and only if it is non-condition operator. Proposition 1.1 Let T be a Schauder operator, then for any invertible operator X, XT is still a Schauder operator. Theorem 1.2 . right Schauder matrix F, suppose G * is its left inverse, then there FPkG * (?) I Theorem 1.3. matrix F is Schauder matrix if and only if it satisfies the following conditions: 1.F have left inverse G *; 2 . FPkG * (?) I. At this point matrix G is a Schauder Theorem 1.4 If for any unitary operator U, FU are Schauder matrix, then F must be a non-conditional matrix in other words, it corresponds to a column vector unconditioned group. Corollary 1.1 Assume Schauder matrix F is a column vector corresponding to the conditions of the base, then there is an ONB {ek} k = 0 ∞ such that {Tek} k = 0 ∞ is not a Schauder basis. Corollary 1.2. Suppose the operator T is generating, then there must be a ONB (?) makes (?) is not a Schauder basis so that we illustrate an article about Olevskii [12] review MR0318848 wrong.

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Functional Analysis > Hilbert space and linear operator theory
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