| Summary: | 碩士 === 中原大學 === 數學研究所 === 86 === We kwon that if A belongs to Mn(C), then the spectral radius of A < 1 if and only if A^{k} converges to 0 (when k converges to infinite). Then we maywant to conjecture that it still keep this property if we have two complex matrices, or a finite set or an infinite set? However, in fact, it would not keep this property for an infinite set. In 1992, Daubechies-Lagarias [2] gave an example. There are two different generalized spectral radius of a set: (1) the joint spectral radius was introduced by Rota-Strang (1960);(2) the generalized spectral radius was introduced by Daubechies-Lagarias [2] (1992). If the set is bounded, then the '' limsup '' in the definition is actually a '' limit ''. The joint spectral radius does not depend on the choice of a norm (as all norms are equivalent on C^{n}). In studying the smoothness properties of compactly and supported wavelets and solutions of two-scale dilation equations, Daubechies-Lagarias conjectured that if a finite set of n-by-n real matrices, then generalized spectral radius is equal to joint spectal radius. Let us make that Berger-Wang's proof is of algebraic-analytic nature and Berger-Wang's method can not be extended to a bounded set of n-by-n complex matrices. Recently, however, Elsner [4] gave an analytic-geometric proof of Daubechies-Lagarias's conjecture (or D-L's conjecture) for a bounded set of n-by-n complex matrices, and Shih [5] gave an analytic-combinatorial proof of D-L's conjecture for a bounded set of n-by-n complex matrices. Thus we have the following (1) Berger-Wang (1992): algebraic-analytic proof of D-L's conjecture; (2) Elsner (1995): analytic-geometric proof of D-L's conjecture; (3) Shih (1998): analytic-combinatorial proof of D-L's conjecture. The purpose of this thesis is to present Elsner's proof and Shih's proof.
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