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The Weyl functional calculus and two-by-two selfadjoint matrices

Published online by Cambridge University Press:  17 April 2009

Werner J. Ricker
Werner J. Ricker
Affiliation:
School of Mathematics, The University of New South Wales, Sydney NSW 2052, Australia
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Let D be a (2 × 2) matrix with distinct eigenvalues λ1 and λ2. There is a basic and well known functional equation which provides a formula for constructing the matrix g (D), for any ℂ-valued function g defined on a subset of ℂ containing {λ1,λ2}, namely .

This equation is used to give a direct and transparent proof of the following fact due to Anderson: A pair of (2 × 2) selfadjoint matrices A1 and A2 commute if and only if the Weyl functional calculus of the pair (A1,A2), which is a matrix-valued distribution, has order zero (that is, is a measure).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 55 , Issue 2 , April 1997 , pp. 321 - 325
Copyright
Copyright © Australian Mathematical Society 1997

References

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