TY - JOUR

T1 - Threshold approximations for the resolvent of a polynomial nonnegative operator pencil

AU - Sloushch, V. A.

AU - Suslina, T. A.

N1 - Publisher Copyright: © 2022 American Mathematical Society

PY - 2022/3/4

Y1 - 2022/3/4

N2 - In a Hilbert space (Formula Presented), a family of operators A(t), t ∈ R, is treated admitting a factorization of the form A(t) = X(t)*X(t), where X(t) = X 0+X 1t+· · ·+X pt p, p ≥ 2. It is assumed that the point λ 0 = 0 is an isolated eigenvalue of finite multiplicity for A(0). Let F(t) be the spectral projection of A(t) for the interval [0, δ]. For |t| ≤ t 0, approximation in the operator norm in (Formula Presented) for the projection F(t) with an error O(t 2p) is obtained as well as approximation for the operator A(t)F(t) with an error O(t 4p) (the so-called threshold approximations). The parameters δ and t 0 are controlled explicitly. Using threshold approximations, approximation in the operator norm in (Formula Presented) is found for the resolvent (A(t) + ε 2pI)−1 for |t| ≤ t 0 and small ε > 0 with an error O(1). All approximations mentioned above are given in terms of the spectral characteristics of the operator A(t) near the bottom of the spectrum. The results are aimed at application to homogenization problems for periodic differential operators in the small period limit

AB - In a Hilbert space (Formula Presented), a family of operators A(t), t ∈ R, is treated admitting a factorization of the form A(t) = X(t)*X(t), where X(t) = X 0+X 1t+· · ·+X pt p, p ≥ 2. It is assumed that the point λ 0 = 0 is an isolated eigenvalue of finite multiplicity for A(0). Let F(t) be the spectral projection of A(t) for the interval [0, δ]. For |t| ≤ t 0, approximation in the operator norm in (Formula Presented) for the projection F(t) with an error O(t 2p) is obtained as well as approximation for the operator A(t)F(t) with an error O(t 4p) (the so-called threshold approximations). The parameters δ and t 0 are controlled explicitly. Using threshold approximations, approximation in the operator norm in (Formula Presented) is found for the resolvent (A(t) + ε 2pI)−1 for |t| ≤ t 0 and small ε > 0 with an error O(1). All approximations mentioned above are given in terms of the spectral characteristics of the operator A(t) near the bottom of the spectrum. The results are aimed at application to homogenization problems for periodic differential operators in the small period limit

KW - Analytic perturbation theory

KW - Correctors

KW - Homogenization theory

KW - Polynomial operator pencils

KW - Threshold approximations

UR - https://www.ams.org/journals/spmj/2022-33-02/S1061-0022-2022-01704-5/

UR - http://www.scopus.com/inward/record.url?scp=85099253239&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/0b0d9e47-7c9e-3315-8b64-410eb74c923b/

U2 - 10.1090/spmj/1704

DO - 10.1090/spmj/1704

M3 - Article

VL - 33

SP - 355

EP - 385

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 2

ER -