Threshold approximations for the resolvent of a polynomial nonnegative operator pencil

Abstract

In a Hilbert space H \mathfrak {H} , a family of operators A ( t ) A(t) , t ∈ R t\in \mathbb {R} , is treated admitting a factorization of the form A ( t ) = X ( t ) ∗ X ( t ) A(t) = X(t)^* X(t) , where X ( t ) = X 0 + X 1 t + ⋯ + X p t p X(t)=X_0+X_1t+\cdots +X_pt^p , p ≥ 2 p\ge 2 . It is assumed that the point λ 0 = 0 \lambda _0=0 is an isolated eigenvalue of finite multiplicity for A ( 0 ) A(0) . Let F ( t ) F(t) be the spectral projection of A ( t ) A(t) for the interval [ 0 , δ ] [0,\delta ] . For | t | ≤ t 0 |t| \le t^0 , approximation in the operator norm in H \mathfrak {H} for the projection F ( t ) F(t) with an error O ( t 2 p ) O(t^{2p}) is obtained as well as approximation for the operator A ( t ) F ( t ) A(t)F(t) with an error O ( t 4 p ) O(t^{4p}) (the so-called threshold approximations). The parameters δ \delta and t 0 t^0 are controlled explicitly. Using threshold approximations, approximation in the operator norm in H \mathfrak {H} is found for the resolvent ( A ( t ) + ε 2 p I ) − 1 (A(t)+\varepsilon ^{2p}I)^{-1} for | t | ≤ t 0 |t|\le t^0 and small ε > 0 \varepsilon >0 with an error O ( 1 ) O(1) . All approximations mentioned above are given in terms of the spectral characteristics of the operator A ( t ) 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.