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On completely irreducible operators
JIANG Zejian, SUN Shanli
Front. Math. China. 2006, 1 (4): 569-581.
https://doi.org/10.1007/s11464-006-0028-4
Let T be a bounded linear operator on Hilbert space H, M an invariant subspace of T. If there exists another invariant subspace N of T such that H = M + N and M ∩ N = 0, then M is said to be a completely reduced subspace of T. If T has a nontrivial completely reduced subspace, then T is said to be completely reducible; otherwise T is said to be completely irreducible. In the present paper we briefly sum up works on completely irreducible operators that have been done by the Functional Analysis Seminar of Jilin University in the past ten years and more. The paper contains four sections. In section 1 the background of completely irreducible operators is given in detail. Section 2 shows which operator in some well-known classes of operators, for example, weighted shifts, Toeplitz operators, etc., is completely irreducible. In section 3 it is proved that every bounded linear operator on the Hilbert space can be approximated by the finite direct sum of completely irreducible operators. It is clear that a completely irreducible operator is a rather suitable analogue of Jordan blocks in L(H), the set of all bounded linear operators on Hilbert space H. In section 4 several questions concerning completely irreducible operators are discussed and it is shown that some properties of completely irreducible operators are different from properties of unicellular operators.
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Orthogonal two-direction multiscaling functions
XIE Changzhen, XIE Changzhen, YANG Shouzhi, YANG Shouzhi
Front. Math. China. 2006, 1 (4): 604-611.
https://doi.org/10.1007/s11464-006-0031-9
The concept of a two-direction multiscaling functions is introduced. We investigate the existence of solutions of the two-direction matrix refinable equation where r × r matrices { P+k } and { P-k } are called the positive-direction and negative-direction masks, respectively. Necessary and sufficient conditions that the above two-direction matrix refinable equation has a compactly supported distributional solution are established. The definition of orthogonal two-direction multiscaling function is presented, and the orthogonality criteria for two-direction multiscaling function is established. An algorithm for constructing a class of two-direction multiscaling functions is obtained. In addition, the relation of both orthogonal two-direction multiscaling function and orthogonal multiscaling function is discussed. Finally, construction examples are given.
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