Toeplitz operators on the polydisk
Toeplitz operators on the polydisk
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DOI：
10.1090/s0002-9939-96-03425-9
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发表时间：
1996
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通讯作者：
Sunhua Sun;Dechao Zheng
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文献类型：
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作者：
Sunhua Sun;Dechao Zheng
本文证明了两个解析Toeplitz算子本质上是双重交换的当且仅当它们在多圆盘的Bergman空间上进行双重交换。设D是C中的开单位圆盘，其边界是圆T。多圆盘Dn和环面T‘是C’的子集，它们分别是n个副本D和T的笛卡儿积。设da(Z)是Dn上的归一化体积度量。Bergman空间L 2(Din)是L2(Dnn，da)的子空间，其函数在Dn中是全纯的。有一个从L2(D，da)到L2(Dn)的正交投影P。L‘(Dn)中符号为f的Toeplitz算子定义为Tf(H)=P(Fh)，对所有h E L 2(Dn)。如果TTG TgTf是紧的，则两个解析Toeplitz算子Tf和Tg本质上是双交换的。如果TTG TgT；=0，则称它们是双重通勤。多圆盘Dn上的函数理论与单位圆盘[R]上的函数理论有很大不同。人们可以预料到，在多面体上的Bergman空间和圆盘上的Bergman空间的算符理论上应该有一些不同。在这篇文章中，我们将证明两个解析Toeplitz算子本质上是双重交换的当且仅当它们在多面体上双重交换。但这在磁盘[AG]、[Z]上是假的。当n>；1时，T‘只是Dn的边界的一小部分，但它是Dn的一个重要部分，被称为Dn的区别边界.Tn也是紧群(以分部乘法为群运算)，因此带有Haar测度.其对偶群为Zn，其中Z为整数群。与[Sw]中一样，我们考虑n维环Tn上的多重傅立叶级数。N环面Tn上的多重傅里叶级数可以看作是L1(Tn)上的傅里叶变换。对于L1(Tin)中的f，其傅里叶变换为f(M)=j f(x，…，Xn)e%(7 X)da(Xi)…Du(Xn)Tn其中m‘=(m1，...，Mn)Ezn和(mf，x)==L mixi，ui(Xi)是T上i=1，...，n的归一化Haar测度.根据[Sw]中的定理1.7，傅立叶变换是内射的，即如果对所有的m E，有f Ell(Tn)且f(M)=0，则f 0.本文的主要结果是如下定理。由编辑于1994年10月6日收到，并经修订后于1995年4月21日收到。1991年数学科目分类。初级47B35。
In this paper it is shown that two analytic Toeplitz operators essentially doubly commute if and only if they doubly commute on the Bergman space of the polydisk. Let D be the open unit disk in C. Its boundary is the circle T. The polydisk Dn and the torus T' are the subsets of C' which are cartesian products of n copies D and T, respectively. Let dA(z) be the normalized volume measure on Dn. The Bergman space L 2(Din) is the subspace of L2(Dnn, dA) whose functions are holomorphic in Dn. There is an orthogonal projection P from L2 (D , dA) onto L2(Dn). The Toeplitz operator with symbol f in L'(D n) is defined by Tf(h) = P(f h), for all h E L 2(D n). Two analytic Toeplitz operators Tf and Tg are said to be essentially doubly commuting if TTg TgTf is compact. They are said to be doubly commuting if TTg TgT; = 0. The function theory on the polydisk Dn is quite different from the function theory on the unit disk [R]. One may expect that there should exist some differences in operator theory on the Bergman spaces between on the polydisk and on the disk. In this paper we will show that two analytic Toeplitz operators essentially doubly commute if and only if they doubly commute on the polydisk. But this is false on the disk [AG], [Z]. Observe that T' is only a small part of the boundary of DTn if n > 1. But it is an important part and is called the distinguished boundary of D n. Tn is also a compact group (with componentwise multiplication as group operation) and as such carries a Haar measure. Its dual group is Zn where Z is the integer group. As in [SW] we consider multiple Fourier series on the n-torus Tn. The multiple Fourier series on the n-torus Tn can be viewed as the Fourier transformation on L1 (Tn). For f in L1 (Tin) the Fourier transformation is given by f(m) = j f (x,... , Xn)e%(7 x) da(xi ) ... du(xn) Tn where m' = (ml,... , mn) E zn and (mf, x) = =l mixi and ui(xi) is the normalized Haar measure on T for i = 1,...,n. By Theorem 1.7 in [SW], the Fourier transformation is injective, i.e. if f E Ll(Tn) and f(m) = 0 for all m E Zn, then f 0_ . The main result in this paper is the following theorem. Received by the editors October 6, 1994 and, in revised form, April 21, 1995. 1991 Mathematics Subject Classification. Primary 47B35.