Abstract
In this chapter we study upper bounds on singular values and determinants of certain operators related to P δ. The bounds are not probabilistic; they only depend on a certain smallness of the perturbation.
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Notes
- 1.
Equation (8.4.7) extends to the case when A 0 and A 1 are trace-class operators as in Sect. 8.4 and the identity is valid for finite-rank perturbations of the identity. By Taylor expansion and partitions of unity we can approximate a C 1 family A t of trace-class perturbations of the identity by a sequence of such perturbations \(A_t^{(\nu )}\) so that \(A_t^{(\nu )}\to A_t\) uniformly in C 1 and similarly for the derivatives (on any given compact interval), and so that \(\mathcal {N} (A_t^{(\nu )})^\perp \cup \mathcal {R}(A_t^{(\nu )})\subset \mathcal {H}^{(\nu )}\), where \(\mathcal {H}^{(\nu )}\) is independent of t and of finite dimension. It then suffices to pass to the limit.
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Sjöstrand, J. (2019). Proof I: Upper Bounds. In: Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations. Pseudo-Differential Operators, vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10819-9_16
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