解析Toeplitz代数的本质换位及其相关问题

在本文中，我们决定出多复变Ｈａｒｄｙ空间Ｈ２上解析Ｔｏｅｐｌｉｔｚ代数的本质换位．即一个算子与所有解析Ｔｏｅｐｌｉｔｚ算子本质可换，当且仅当它是符号属于Ａｃ的Ｔｏｅｐｌｉｔｚ算子的紧扰动．由此，符号属于Ａｃ的Ｔｏｅｐｌｉｔｚ算子生成的代数Ｆ（Ａｃ）在Ｃａｌｋｉｎ代数中的像是极大可换闭代数，这导致了Ｌ．Ｃｏｂｕｒｎ正合列的极大扩充．从这个事实，证明了符号属于Ａｃ的Ｔｏｅｐｌｉｔｚ算子的本质谱是连通的，这大大改进了Ｃ－Ｓ最近的工作．从本文的主要定理，证明了Ｔｏｅｐｌｉｔｚ代数Ｆ（Ｌ∞）的本质换位和本质中心是由符号属于ＱＣ的Ｔｏｅｐｌｉｔｚ算子生成的代数Ｆ（ＱＣ），这些结果又导致了对代数Ｆ（Ｈ∞）＋Ｋ自同构群的确定．

The Essential Commutant of Analytic Toeplitz Algebra and Their Some RelatedProblems

In this paper,the essential commutant of analytic Toeplitz algebra F(H∞) on Hardy space H2 on the unit sphere Sn of Cn is determined. An operator commutes essentially with all analytic Toeplitz operators if and only if it is a compact perturbation of a Toeplitz operator with symbol in Ac It follows that the image in the Calkin algebra of the algebra F(Ac) generated by Toeplitz operators with symbol in Ac is a maximal abelian algebra, which leads to a maximal expansion of Coburn's exact sequence. From this fact, we prove that the essential spectrum of Toeplitz operators with symbol in Ac are connected, which extends the recent results of C-S.We obtain also that the essential commutant and center of Toeplitz algebra F(L∞)are the algebra generated by Toeplitz operators with symbol in QC by the main theorem in this paper,These results lead to a characterization of automorphisms of the algebra F(H∞) + κ.