Weyl–von Neumann theorem
In mathematics, the Weyl–von Neumann theorem is a result in operator theory due to Hermann Weyl and John von Neumann. It states that, after the addition of a compact operator (Weyl (1909)) or Hilbert–Schmidt operator (von Neumann (1935)) of arbitrarily small norm, a bounded self-adjoint operator or unitary operator on a Hilbert space is conjugate by a unitary operator to a diagonal operator. The results are subsumed in later generalizations for bounded normal operators due to David Berg (1971, compact perturbation) and Dan-Virgil Voiculescu (1979, Hilbert–Schmidt perturbation). The theorem and its generalizations were one of the starting points of operator K-homology, developed first by Lawrence G. Brown, Ronald Douglas and and, in greater generality, by .
In 1958 Kuroda showed that the Weyl–von Neumann theorem is also true if the Hilbert–Schmidt class is replaced by any Schatten class Sp with p ≠ 1. For S1, the trace-class operators, the situation is quite different. The Kato–Rosenblum theorem, proved in 1957 using scattering theory, states that if two bounded self-adjoint operators differ by a trace-class operator, then their absolutely continuous parts are unitarily equivalent. In particular if a self-adjoint operator has absolutely continuous spectrum, no perturbation of it by a trace-class operator can be unitarily equivalent to a diagonal operator.
References[]
- Conway, John B. (2000), A Course in Operator Theory, Graduate Studies in Mathematics, American Mathematical Society, ISBN 0821820656
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- Higson, Nigel; Roe, John (2000), Analytic K-Homology, Oxford University Press, ISBN 0198511760
- Katō, Tosio (1995), Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, vol. 132 (2nd ed.), Springer, ISBN 354058661X
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- von Neumann, John (1935), Charakterisierung des Spektrums eines Integraloperators, Actualités Sci. Indust., vol. 229, Hermann
- Weyl, Hermann (1909), "Über beschränkte quadratische Formen, deren Differenz vollstetig ist" (PDF), Rend. Circolo Mat. Palermo, 27: 373–392, doi:10.1007/bf03019655
- Operator theory
- Theorems in functional analysis
- K-theory
- Mathematical analysis stubs