A Matrix-Valued Aleksandrov Disintegration Theorem

作者：Elliott Sam^*

来源：Complex Analysis and Operator Theory, 2010, 4(2): 145-157.

DOI：10.1007/s11785-009-0007-3

• Aleksandrov-Clark measure
• Matrix-valued measure
• Disintegration Theorem

The system of measures constructed by Clark (J Anal Math 25: 169-191, 1972), and studied extensively by Aleksandrov and others provide useful information about the properties of analytic self-maps of the unit disc. In their paper, Kapustin and Poltoratski (J Funct Anal 238: 313-326, 2006) extended the concept to include operator (and hence matrix) valued analytic functions, though they restrict their attention to the interesting special case of operator-valued inner functions. In this paper, we present an alternative (though equivalent) construction, which applies only to the matrix-valued case, though not only to inner functions. This construction provides us with interesting analogues of certain theorems known to be true in the scalar case. In particular, we present a proof of a matrix-valued generalisation of the celebrated Aleksandrov Disintegration Theorem (Inv Akad Nauk ArmSSR Mat 22: 490-503, 1987) to a certain class of matrix-valued functions.

• 出版日期2010-5