Schauder Basis And Operator Theory

In the study of Banach space and Hilbert space, the concept of basis is a very useful and very important tool. Since the definition of Schauder basis was proposed by J.schauder in 1927, there have been a large number of articles and books about Schauder basis. And now Schauder basis has become a specialized field. However,the combinational research of operator theory and Schauder basis is very rare. An article of Olevskii "On Operators Generating Condition Basis in a Hilbert Spaces" is a masterpiece on the combination of operator theory and Schauder basis. English translation of this article made a serious mistake. However, the professionals who review this article actually repeat this mistake word for word. That still appear in the literature review. The following are comments entry:MR0318848 (47#7394) Olevskii, A. M. Operators that produce conditional basis in a Hilbert space. (Russian) Mat. Zametki 12 (1972),73 84. (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.In the second section, we introduce some basic definitions and results about the exis-tence of Schauder basis, the existence of Schauder basic sequence, the unconditional basis and the unconditional basic sequence.The third, fourth and fifth sections are the main content of this thesis. In this part we try to do the following two things on the mistake about Olevskii's article.1. Point out the mistake, give the corresponding proof and detailed characterization.2. Collect the relevant results, introduce the core theorem in Olevskii's article.Definition 1.1. A bounded linear operator T€B(H) called Schauder operator, if there is some ONB{(?), s.t. the column vector of oo x oo matrix of T on{(?) form a Schauder basic sequence. Definition 1.2. A∞×∞matrix called Schauder matrix, if its column vector{(?) form a Schauder basic sequence.Theorem 1.1. A compact operator K is Schauder operator if and only if it is unconditional 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. For Schauder matrix F, let G* is its left inverse matrix, then FPkG*(?)I.Theorem 1.3. Matrix F is a Schauder matrix if and only if the following conditions are hold:1. F has the left inverse matrix G*:2. FPkG*(?)I. At this time,G is a Schauder matrix.Theorem 1.4. If for any unitary operator U, FU is Schauder matrix, then F must be a unconditional matrix. In another words,column vector of F form a unconditional basis.Corollary 1.1. Suppose the column vector of Schauder matrix F form a condition basis, then there must be a ONB{(?),s.t.{Tek}(?) is not a Schauder basis.Corollary 1.2. Suppose T is a generating operator, then there must be a ONB{(?),s.t. {(?) is not a Schauder basis.Thus we note that the comments MR0318848 on the article of Olevskii[12] is wrong.