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Bounds on the spectrum and reducing subspaces of a \(J\)-self-adjoint operator. (English) Zbl 1236.47032

Let \(A_0\) and \(A_1\) be selfadjoint operators in some Hilbert spaces \(H_+\) and \(H_-\), respectively, and define the operators \(J:= I \oplus -I\) and \(A:=A_0 \oplus A_1\) in the Hilbert space \(H:=H_+\oplus H_-\). Let \(V\) be a bounded \(J\)-selfadjoint operator in \(H\) which anti-commutes with \(J\). Then the operator \[ L:=A+V \] is \(J\)-selfadjoint in the Krein space \((H, [\cdot,\cdot])\) with \([\cdot,\cdot]:=(J\cdot,\cdot)\). Here, \(L\) is considered as the perturbed operator, whereas the unperturbed operator is \(A\) which commutes with the fundamental symmetry \(J\) and has \(H_+\) and \(H_-\) as reducing subspaces. Such operators are sometimes called fundamental reducible. Under the additional assumption that there exists a maximal uniformly positive reducing subspace \(H_+^\prime\) for \(L\), bounds on the position of this subspace relative to \(H_+\) are established. Here, the position of one subspace relative to another subspace is measured with the operator angle which is a non-negative operator. In the case that the spectra of \(A_0\) and \(A_1\) are disjoint and have a positive distance \(d>0\), a bound for the operator angle \(\Theta\) between \(H_+\) and \(H_+^\prime\) was already shown in [S.Albeverio, A. K.Motovilov and A. A.Shkalikov, Integral Equations Oper.Theory 64, 455–486 (2009; Zbl 1197.47024)] under the condition \(2\|V\|<d\): \[ \tan \Theta \leq \tanh \left( \frac{1}{2} \text{arctanh}\, \frac{2\|V\|}{d} \right). \] This bound relies on the disjointness of the spectra of \(A_0\) and \(A_1\) of the unperturbed operator \(A\) and involves this distance between the spectra of \(A_0\) and \(A_1\). In this sense, it is an a priori estimate.
In the present paper, these investigations are continued. In general, for these new bounds to hold, the disjointness of the spectra of \(A_0\) and \(A_1\) is not required at all. Instead, it is assumed that the spectra of \(A_0\) and \(L| (H_+^\prime)^{[\perp]}\) or the spectra of \(A_1\) and \(L| H_+^\prime\) are disjoint. Moreover, a so-called a posteriori estimate is proven which involves only the distance between the spectrum of \(L| H_+^\prime\) and \(L| (H_+^\prime)^{[\perp]}\). The obtained bounds represent analogues of the celebrated trigonometric estimates for selfadjoint operators known as Davis-Kahan \(\sin \Theta\), \(\sin 2\Theta\), \(\tan \Theta\) and \(\tan 2\Theta\) theorems. As an example, the quantum harmonic oscillator under a \(\mathcal{PT}\)-symmetric perturbation is discussed.

MSC:

47B50 Linear operators on spaces with an indefinite metric
47A10 Spectrum, resolvent
47A55 Perturbation theory of linear operators
47A62 Equations involving linear operators, with operator unknowns

Citations:

Zbl 1197.47024
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