Fredholm composition operators
Fredholm composition operators
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DOI：
10.1090/s0002-9939-1980-0565345-0
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发表时间：
1980-02
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影响因子：
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通讯作者：
Ashok Kumar
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文献类型：
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作者：
Ashok Kumar
本文给出了L2[0，1]上复合算子CT是Fredholm算子的一个充要条件.此外，刻画了12（N）上的所有Fredholm复合算子. 1.你的助手。设（X，S，X）是a-有限测度空间，T是从X到其自身的可测非奇异变换（当X（E）= 0时AT-'（E）= 0）.然后，L2（X，S，X）上的复合变换CT被定义为对于每个EL 2（X，S，X），CT J = foT。如果CT是值域在L2（X，，A）中的有界算子，我们称它为由T诱导的复合算子。本文主要研究L2（X，S，X）（简称L2（X））上的Fredhohm复合算子，其中X是单位区间，S是X的所有Borel子集的α-代数，X是S上的Lebesgue测度。本文还给出了复合算子是12（N）上Fredholm的一个判别准则.令B（L2（X））、R（CT）1和[x，y，z，. ]表示L ~ 2（X）上所有有界线性算子的Banach代数，CT值域的正交补和向量x，y，z，. . .分别定义.希尔伯特空间H上的算子A称为Fredholm算子，如果A的值域是闭的，并且核和上核的维数是有限的。2. Fredholm复合算子J. Cima，J. Thomson和W. Wogen在[2]中证明了复合算子CT是Fredholm当且仅当T是圆盘的共形自同构。下面的定理给出了L2（X）= L2[0，1]上Fredholm复合算子的类似特征。定理1.设CT E B（L2（X））.则CT是Fredholm算子当且仅当它是可逆的。证据如果CT是可逆的，那么CT显然是一个Fredholm算子。由[5，p.82]可知CTCT = Mf 0，其中Mf是由fo = dAT-/dA导出的乘法算子。171-174]和Ker CT = Ker CTCT = Ker Mf= L2（X 0），其中X 0 = {x：f0（x）= 0），由此得出CT的核的维数为零或无穷大。编辑于1977年5月3日收到，修订版于1979年6月18日收到。AMS（MOS）主题分类（1970年）。小学47 B30;中学47 B30。
In this paper a necessary and sufficient condition for a composition operator CT on L2[0, 1] to be a Fredholm operator is given. In addition, all Fredholm composition operators on 12(N) are characterized. 1. Preliminaries. Let (X, S, X) be a a-finite measure space and T be a measurable nonsingular (AT-'(E) = 0 whenever X(E) = 0) transformation from X into itself. Then a composition transformation CT on L2(X, S, X) is defined as CTJ = f o T for everyf EL2(X, ,X ). In case CT is a bounded operator with range in L2(X, , A), we call it a composition operator induced by T. The main purpose of this paper is to study Fredhohm composition operators on L2(X, S, X) (briefly written as L2(X)), where X is the unit interval, S is the a-algebra of all Borel subsets of X, and X is the Lebesgue measure on S. A criterion for a composition operator to be Fredholm on 12(N) is also given here. Let B(L2(X)), R(CT)1 and [x,y, z,... ] denote the Banach algebra of all bounded linear operators on L2(X), the orthogonal complement of the range of CT and the closed linear span of the vectors x, y, z, . . . respectively. DEFINITION. An operator A on a Hilbert space H is called a Fredholm operator if the range of A is closed and if the dimensions of the kernel and the cokernel are finite. 2. Fredholm composition opertors. A characterization of Fredholm composition operators on H2(D) is given by J. Cima, J. Thomson and W. Wogen in [2] where they proved that a composition operator CT is Fredholm if and only if T is a conformal automorphism of the disc. The following theorem gives an analogous characterization of Fredholm composition operators on L2(X) = L2[0, 1]. THEOREM 1. Let CT E B(L2(X)). Then CT is a Fredholm operator if and only if it is invertible. PROOF. If CT is invertible, then clearly CT is a Fredholm operator. It is known from [5, p. 82] that CTCT = Mf0, where Mf is the multiplication operator induced by fo = dAT-/dA. Since X is nonatomic [3, pp. 171-174] and Ker CT = Ker CTCT = Ker Mf= L2(Xo), where X0 = {x: f0(x) = 0), it follows that the dimension of the kernel of CT of either zero or infinite. Received by the editors May 3, 1977 and, in revised form, June 18, 1979. AMS (MOS) subject classifications (1970). Primary 47B30; Secondary 47B30.