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Divided differences and multivariate holomorphic calculus
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DOI：10.4171/dm/1060.png)
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Abstract
En 中文
We review the multivariate holomorphic functional calculus for tuples in a commutative Banach algebra and establish a simple na & iuml;ve extension to commuting tuples in a general Banach algebra. The approach is na & iuml;ve in the sense that the na & iuml;vely defined joint spectrum maybe too big. The advantage of the approach is that the functional calculus then is given by a simple concrete formula from which all its continuity properties can easily be derived. We apply this framework to multivariate functions arising as divided differences of a univariate function. This provides a rich set of examples to which our na & iuml;ve calculus applies. Foremost, we offer a natural and straightforward proof of the Connes-Moscovici Rearrangement Lemma in the context of the multivariate holomorphic functional calculus. Secondly, we show that the Daletski-Krein type noncommutative Taylor expansion is a natural consequence of our calculus. Also Magnus' Theorem which gives a nonlinear differential equation for the log of the solutions to a linear matrix ODE follows naturally and easily from our calculus. Finally, we collect various combinatorial related formulas.
Keywords:
divided difference
Banach algebra
Rearrangement Lemma
holomorphic functional calculus
Journal
D
IF:
0.6
Papers:
44
Citations:
0
