Diagonalizing operators with reflection symmetry
| dc.creator | Jorgensen, Palle E. T. | |
| dc.date | 1999-08-05 | |
| dc.date.accessioned | 2026-07-07T05:30:12Z | |
| dc.date.available | 2026-07-07T05:30:12Z | |
| dc.description | Let $U$ be an operator in a Hilbert space $\mathcal{H}_{0}$, and let $\mathcal{K}\subset\mathcal{H}_{0}$ be a closed and invariant subspace. Suppose there is a period-2 unitary operator $J$ in $\mathcal{H}_{0}$ such that $JUJ=U^*$, and $PJP \geq 0$, where $P$ denotes the projection of $\mathcal{H}_{0}$ onto $\mathcal{K}$. We show that there is then a Hilbert space $\mathcal{H}(\mathcal{K})$, a contractive operator $W:\mathcal{K}\to\mathcal{H}(\mathcal{K})$, and a selfadjoint operator $S=S(U)$ in $\mathcal{H}(\mathcal{K})$ such that $W^*W=PJP$, $W$ has dense range, and $SW=WUP$. Moreover, given $(\mathcal{K},J)$ with the stated properties, the system $(\mathcal{H}(\mathcal{K}),W,S)$ is unique up to unitary equivalence, and subject to the three conditions in the conclusion. We also provide an operator-theoretic model of this structure where $U|_{\mathcal{K}}$ is a pure shift of infinite multiplicity, and where we show that $\ker(W)=0$. For that case, we describe the spectrum of the selfadjoint operator $S(U)$ in terms of structural properties of $U$. In the model, $U$ will be realized as a unitary scaling operator of the form $f(x)\mapsto f(cx)$, $c>1$, and the spectrum of $S(U_{c})$ is then computed in terms of the given number $c$. | |
| dc.description | 30 pages; Dedicated to the memory of I.E. Segal | |
| dc.identifier | https://arxiv.org/abs/math/9908021 | |
| dc.identifier | http://arxiv.org/abs/math/9908021 | |
| dc.identifier | J. Funct. Anal. 190 (2002), 93-132 | |
| dc.identifier | doi:10.1006/jfan.2001.3881 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/78917 | |
| dc.subject | Functional Analysis | |
| dc.subject | 47A05; 47A66; 47B15 | |
| dc.title | Diagonalizing operators with reflection symmetry | |
| dc.type | text |