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

Xia, Jingbo

Canadian mathematical bulletin, Tome 47 (2004) no. 1, pp. 144-151

Résumé

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 ${{\lim }_{n\,\to \,\infty }}\,{{n}^{-1/2}}\,\left\| {{P}_{n}} \right\|\mathcal{C}\,=\,0$ , 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.)

Détail

DOI : 10.4153/CMB-2004-015-4

Mots-clés : 47A40, 47B10

Bibliographie

[1] [1] Carey, R. and Pincus, J., Unitary equivalence modulo the trace class for self-adjoint operators. Amer. J. Math. 98 (1976), 481–514. Google Scholar

[2] [2] Carey, R. and Pincus, J., Mosiacs, principal functions, and mean motion in von Neumann algebra. Acta Math. 138 (1977), 153–218. Google Scholar

[3] [3] David, G. and Voiculescu, D., s-Numbers of singular integrals for the invariance of absolutely continuous spectra in fractional dimensions. J. Funct. Anal. 94 (1990), 14–26. Google Scholar

[4] [4] Gohberg, I. and Krein, M., Introduction to the theory of linear nonselfadjoint operators. Amer.Math. Soc., Transl. Math.Monogr. 18, Providence, 1969. Google Scholar

[5] [5] Kato, T., Perturbation of continuous spectra by trace class operators. Proc. Japan Acad. 33 (1957), 260–264. Google Scholar

[6] [6] Kato, T., Perturbation theory for linear operators. Springer-Verlag, New York, 1976. Google Scholar

[7] [7] Rosenblum, M., Perturbations of continuous spectrum and unitary equivalence. Pacific J. Math. 7 (1957), 997–1010. Google Scholar

[8] [8] Reed, M. and Simon, B., Methods of modern mathematical physics, III, Scattering theory. Academic Press, New York, 1979. Google Scholar

[9] [9] Voiculescu, D., Some results on norm-ideal perturbations of Hilbert space operators. J. Operator Theory 2 (1979), 3–37. Google Scholar

[10] [10] Voiculescu, D., Some results on norm-ideal perturbations of Hilbert space operators. II. J. Operator Theory 5 (1981), 77–100. Google Scholar

[11] [11] Voiculescu, D., On the existence of quasicentral approximate units relative to normed ideals. Part I. J. Funct. Anal. 91 (1990), 1–36. Google Scholar

[12] [12] Xia, J., An analogue of the Kato-Rosenblum theorem for commuting tuples of self-adjoint operators. Comm. Math. Phys. 198 (1998), 187–197. Google Scholar

[13] [13] Xia, J., Trace-class perturbation and strong convergence: wave operators revisited. Proc. Amer.Math. Soc. 128 (2000), 3519–3522. Google Scholar