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On the Uniqueness of Wave Operators Associated With Non-Trace Class Perturbations |
Canad. Math. Bull. 47(2004), 144-151
Published:2004-03-01
Printed: Mar 2004
Printed: Mar 2004
- Jingbo Xia
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 - 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] |