Research on operators with natural spectrum

自然频谱算子研究

• 批准号: 09640166
• 负责人: HATORI Osamu
• 金额: $ 1.86万
• 依托单位: Niigata University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640166/
• 关键词: commtative Banach algebra; locally compact abelian group; spectrum; Wiener-Pitt phenomenum; Fourier multiplier; operating function; Apostol algebra; decomposable operator; Wiener-Pitt現象; 測度; キーワード1局所コンパクトabel群; キーワード2Wiener-Pitt現象; キーワード3自然なスペ

We gave a sufficient condition for operating functions defined on a certain Banach function space to be only Lipschitz functions. Operating functions on non-trivial Banach function al-gebras or spaces need not be Lipschitz, but sutisifies strong continuity property. This means implicitly that it is hard to characterize the Gelfand space for a Banach function algebra in terms of operating functions.Let M(G) be the measure algebra on a non-discrete locally compact abelian group G and NS(G) denote the set of all measures in M(G) with natura1 spectrum. Then, NS(G) is not closed under addition and NS(G) + L^1(G) = M(G) holds if G is not compact. Let M_0(G) be a closed subalgebra of M(G) which consisit of all measurs whose Fuorier-Stieltjes transforms vanish at infinity and NS_0(G) denotes the subset of M_0(G) whose element have natura1 spectra. If G is compact NS_0(G) coincides with the Apostol algebra of M_0(G), which is not the case for non-compact G.There exists a measure mu in NS(G) such that mu is not decomposable as an operator on L^1(G). In particular, NS(G)+ NS(G)+ NS(G) = M(G) holds.The Apostol algebra of a Douglas algebra coincides with the algebra of all Q-continuous functions. Let H^* be the algebra of all bounded analytic functions on the open unit disk. Then NSH^*+ NSH^* = H^* holds, thus NSH^* is not closed under addition and it is rather large subset of H^*.

给出了定义在某个Banach函数空间上的算子函数仅为Lipschitz函数的一个充分条件。非平凡Banach函数代数或空间上的算子函数不一定是Lipschitz的，但满足强连续性。设M（G）是非离散局部紧交换群G上的测度代数，NS（G）表示M（G）中所有具有自然谱的测度的集合.若G不是紧的，则NS（G）在加法下不闭，且NS（G）+ L^1（G）= M（G）成立.设M_0（G）是M（G）的一个闭子代数，它由其Fuorier-Stieltjes变换在无穷远处为零的所有测度组成，NS_0（G）表示M_0（G）中元素具有自然谱的子集。如果G是紧的，则NS（G）与M_0（G）的Apostol代数重合，而非紧的G则不是这样. NS（G）中存在测度μ，使得μ不能分解为L^1（G）上的算子.特别地，NS（G）+ NS（G）+ NS（G）= M（G）成立，道格拉斯代数的Apostol代数与所有Q-连续函数的代数一致.设H^* 是开单位圆盘上所有有界解析函数的代数。则NSH^*+ NSH^* = H^* 成立，因此NSH^* 在加法下不是闭的，它是H^* 的相当大的子集。

项目成果

O.Hatori: "Non-Lipschitz functions which operate function spaces" Scientiae Mathematicae. 1. 7-14 (1998)

O.Hatori：“操作函数空间的非 Lipschitz 函数”Scientiae Mathematicae。

K.Izuchi: "A_ψ-invariant subspaces on the torus" Canacl.J.Math.50. 99-133 (1998)

K.Izuchi：“环面上的 A_ψ 不变子空间”Canacl.J.Math.50 (1998)。

G.Ji: "Certain invariant Subspaie structure of L^2(T^2)" Proceedings of Amer.Math.Soc.126. 2361-2368 (1998)

G.Ji：“L^2(T^2) 的某些不变 Subspaie 结构”Amer.Math.Soc.126 论文集。

S.-E.Takahasi: "A structure of ring homomorphism on commutative Banach algebras" Proc.Amer.Math.Soc.(to appear).

S.-E.Takahasi：“交换巴纳赫代数上的环同态结构”Proc.Amer.Math.Soc.（即将出现）。

T.Ishii: "BKW-operators for Chebyshev systems" Tokyo.J.Math.(to appear).

T.Ishii：“切比雪夫系统的 BKW 算子”Tokyo.J.Math.（即将出现）。