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Research Papers
Preservation of absolutely continuous spectrum for contractive operators
C. Liawab, S. Treilc a Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA
b CASPER Baylor University, Waco, TX 76798, USA
c Department of Mathematics, Brown University, Providence, RI 02912, USA
Abstract:
Contractive operators $T$ that are trace class perturbations of a unitary operator $U$ are treated. It is proved that the dimension functions of the absolutely continuous spectrum of $T$, $T^* ,$ and of $U$ coincide. In particular, if $U$ has a purely singular spectrum, then the characteristic function $\theta$ of $T$ is a two-sided inner function, i.e., $\theta(\xi)$ is unitary a.e. on $\mathbb{T}$. Some corollaries to this result are related to investigations of the asymptotic stability of the operators $T$ and $T^*$ (the convergence $T^n\to 0$ and $(T^*)^n\to 0$, respectively, in the strong operator topology). The proof is based on an explicit computation of the characteristic function.
Keywords:
Trace class perturbations, contractive operators, dimension function, absolutely continuous spectrum.
Received: 04.10.2021
Citation:
C. Liaw, S. Treil, “Preservation of absolutely continuous spectrum for contractive operators”, Algebra i Analiz, 34:3 (2022), 232–251; St. Petersburg Math. J., 34:3 (2023), 483–496
Linking options:
https://www.mathnet.ru/eng/aa1817 https://www.mathnet.ru/eng/aa/v34/i3/p232
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Abstract page: | 92 | Full-text PDF : | 1 | References: | 17 | First page: | 11 |
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