Some Lemmata on the Perturbation of the Spectrum
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Some Lemmata on the Perturbation of the Spectrum A. I. Nazarov∗,∗∗,1 ∗
St. Petersburg Department of Steklov Institute, Fontanka 27, St.Petersburg, 191023, Russia ∗∗ St. Petersburg State University, Universitetskii pr. 28, St. Petersburg, 198504, Russia, E-mail: 1 [email protected] Received May 16, 2020; Revised May 20, 2020; Accepted June 2, 2020
Abstract. We give some sufficient conditions for the preservation of the second term in the spectral asymptotics of a compact operator under the perturbation of the metrics in the Hilbert space. DOI 10.1134/S1061920820030085
As is well known, see, e.g., [1, Lemma 1.16], the one-term power-type spectral asymptotics of a compact operator in the Hilbert space does not change under a compact perturbation of the metrics of the space. The problem of preserving the two-term asymptotics is much more sensitive and complicated. Here we give some sufficient conditions for that preservation. The results can be applied in the spectral analysis of some integro-differential operators arising in the theory of Gaussian random processes, see [3]. In what follows, we denote by c any absolute constant. Lemma 1. Let K and B be self-adjoint compact operators in the Hilbert space H. Suppose that K and I + B are positive. Denote by λn the eigenvalues of K enumerated in the decreasing order taking into account the multiplicities, and by hn the corresponding normalized eigenfunctions. Finally, suppose that −B , λn = an + b + O(n−δ )
Bhn H cn−(1+δ) ,
as n → ∞, where a, B, δ > 0, b ∈ R. Then the eigenvalues λn of generalized eigenvalue problem Khn = λn hn + Bhn
(1)
(2)
have the same two-term asymptotics as n → ∞: −B . λn = an + b + O(n−δ )
(3)
Proof. We introduce a new inner product in H: h, g := (h + Bh, g)H . 1
It is easy to see that the corresponding norm |||h||| := h, h 2 is equivalent to the original one. Denote by H the space H with the new inner product. Then the sesquilinear form (Kh, g)H generates a compact positive self-adjoint operator K such that Kh, g = (Kh, g)H ,
h, g ∈ H,
and the generalized eigenvalue problem (2) is reduced to the standard eigenvalue problem for the operator K in H. Recall some elementary facts from the theory of spectral measure, see, e.g., [2, Chap. 5]. The spectral measure dE(t) associated with K generates the family of scalar measures deh (t) := dE(t)h, h,
h ∈ H.
Moreover, the following obvious formulas hold for arbitrary h ∈ H: |||h|||2 = deh (t), |||Kh − λh|||2 = (t − λ)2 deh (t). R
R
378
SOME LEMMATA ON THE PERTURBATION OF THE SPECTRUM
379
If we assume that an interval Δ = (λ − κ, λ + κ) is free from the spectrum of K, then, for any h ∈ H, 2 2 2 |||Kh − λh||| = (t − λ) deh (t) κ deh (t) = κ 2 |||h|||2 . (4) R\Δ
R\Δ
Now we set λ = λn , h = hn . For any g ∈ H, |Khn − λn hn , g| = |(Khn , g)H − λn (hn + Bhn , g)H | = λn |(Bhn , g)H | λn BhnH gH
c1 λn |||g|||, n1+δ
and therefore
c1 c2 λn 1+δ λn |||hn |||. n1+δ n Comparing this inequality with (4), we see that
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