Segal–Bargmann Transform, Functional Calculus on Matrix Spaces and the Theory of Semi-circular and Circular Systems
Segal–Bargmann Transform, Functional Calculus on Matrix Spaces and the Theory of Semi-circular and Circular Systems
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DOI：
10.1006/jfan.1996.2990
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发表时间：
1997-02
影响因子：
1.7
通讯作者：
P. Biane
中科院分区：
文献类型：
--
作者：
P. Biane
设Md(C)为d×d复矩阵空间，Xd为Hermite矩阵的子空间。研究了Xd上取值于Md(C)的函数的Segal-Bargmann变换，这些变换是由函数演算给出的。我们证明了当→∞时，这种映射的变换变得接近于Md(C)上的全纯泛函演算的空间，并且在极限上产生了Wigner半圆分布的L2空间和圆盘的Hardy空间之间的等距.我们将这一等距与Voulescu的圆形和半圆形系统理论联系起来，并研究了在高维酉群上用霍尔变换代替西格尔-巴格曼变换时的等距关系。
LetMd(C) be the space ofd×dcomplex matrices, and Xdbe the subspace of hermitian matrices. We study the Segal–Bargmann transform of functions on Xd, with values inMd(C), which are given by functional calculus. We show that whend→∞, the transform of such a map becomes close to the space of holomorphic functional calculus onMd(C), and that this yields, in the limit, an isometry between theL2space of Wigner's semi-circle distribution and the Hardy space of the disk. We relate this isometry to Voiculescu's theory of circular and semi-circular systems, and we study its analogue when the Segal–Bargmann transform is replaced by the Hall transform on unitary groups of large dimensions.