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On the Uniqueness of Wave Operators Associated With Non-Trace Class Perturbations

  Published:2004-03-01
 Printed: Mar 2004
  • Jingbo Xia
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Abstract

Voiculescu has previously established the uniqueness of the wave operator for the problem of $\mathcal{C}^{(0)}$-perturbation of commuting tuples of self-adjoint operators in the case where the norm ideal $\mathcal{C}$ has the property $\lim_{n\rightarrow\infty} n^{-1/2}\|P_n\|_{\mathcal{C}}=0$, where $\{P_n\}$ is any sequence of orthogonal projections with $\rank(P_n)=n$. We prove that the same uniqueness result holds true so long as $\mathcal{C}$ is not the trace class. (It is well known that there is no such uniqueness in the case of trace-class perturbation.)
MSC Classifications: 47A40, 47B10 show english descriptions Scattering theory [See also 34L25, 35P25, 37K15, 58J50, 81Uxx]
Operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20]
47A40 - Scattering theory [See also 34L25, 35P25, 37K15, 58J50, 81Uxx]
47B10 - Operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20]
 

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