Compact Operators via the Berezin Transform
Compact Operators via the Berezin Transform
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DOI：
10.1512/iumj.1998.47.1407
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发表时间：
1998-07
影响因子：
1.1
通讯作者：
S. Axler;Dechao Zheng
中科院分区：
文献类型：
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作者：
S. Axler;Dechao Zheng
本文证明了如果S等于单位圆盘上Bergman空间上Toeplitz算子的fi乘积的fi和，则E是紧的当且仅当∂D上的S的Berezin变换等于0。即使当S等于单个Toeplitz算子时，这一结果也是新的。我们的主要结果可以用Unifi的方法证明几个已知的关于紧Toeplitz算子、紧Hankel算子及其适当乘积的结果。1导言设da表示单位圆盘D上的勒贝格面积测度，使D的测度等于1。Bergman空间L 2a是由D上的解析函数组成的Hilbert空间，这些函数也在L 2(D，da)中。对z∈D，Bergman再生核是函数K z∈L 2a，使得f(Z)=Hf，Kz I对任意f∈L 2a。归一化Bergman再生核kz是函数Kz/kkzk2。这里，和本文的其他地方一样，范数k k 2和内积h，i取在空间L 2(D，da)上。对S来说，L上的一个有界算子
AbstractIn this paper we prove that if S equals a finite sum of finite productsof Toeplitz operators on the Bergman space of the unit disk, thenS is compact if and only if the Berezin transform of S equals 0 on∂D. This result is new even when S equals a single Toeplitz operator.Our main result can be used to prove, via a unified approach, severalpreviously known results about compact Toeplitz operators, compactHankel operators, and appropriate products of these operators. 1 Introduction Let dA denote Lebesgue area measure on the unit disk D, normalizedso that the measure of D equals 1. The Bergman space L 2a is the Hilbertspace consisting of the analytic functions on D that are also in L 2 (D,dA).For z ∈ D, the Bergman reproducing kernel is the function K z ∈ L 2a suchthatf(z) = hf,K z ifor every f ∈ L 2a . The normalized Bergman reproducing kernel k z is thefunction K z /kK z k 2 . Here, as elsewhere in this paper, the norm k k 2 and theinner product h , i are taken in the space L 2 (D,dA).For S a bounded operator on L