Abstract
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.
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