TY - JOUR

T1 - Principal part of resolvent and factorization of an increasing nonanalytic operator-function

AU - Markus, Alexander

AU - Matsaev, Vladimir

PY - 1991/9/1

Y1 - 1991/9/1

N2 - Let a selfadjoint operator-valued function L(λ) be given on the interval [a,b] such that L(a)≪0, L(b)≫0, L′(λ)≫0 (a≤λ≤b), and L″(λ) has a certain smoothness (for instance, it satisfies Hölder's condition). It turns out that the spectral theory of the operator-valued function L(λ) can be reduced to the spectral theory of one operator Z, the spectrum of which lies on (a, b) and which is similar to a selfadjoint operator. In particular, the factorization takes place:L(λ)=M(λ)(λI-Z), where the operator-valued function M(λ) is invertible on [a, b]. Earlier similar results were known only for analytic operator-valued functions. The authors had to use new methods for the proof of the described theorem. The key moment is the decomposition of L-1(λ) into the sume of its principal and regular parts.

AB - Let a selfadjoint operator-valued function L(λ) be given on the interval [a,b] such that L(a)≪0, L(b)≫0, L′(λ)≫0 (a≤λ≤b), and L″(λ) has a certain smoothness (for instance, it satisfies Hölder's condition). It turns out that the spectral theory of the operator-valued function L(λ) can be reduced to the spectral theory of one operator Z, the spectrum of which lies on (a, b) and which is similar to a selfadjoint operator. In particular, the factorization takes place:L(λ)=M(λ)(λI-Z), where the operator-valued function M(λ) is invertible on [a, b]. Earlier similar results were known only for analytic operator-valued functions. The authors had to use new methods for the proof of the described theorem. The key moment is the decomposition of L-1(λ) into the sume of its principal and regular parts.

UR - http://www.scopus.com/inward/record.url?scp=0040995915&partnerID=8YFLogxK

U2 - 10.1007/BF01200556

DO - 10.1007/BF01200556

M3 - Article

AN - SCOPUS:0040995915

VL - 14

SP - 716

EP - 746

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

SN - 0378-620X

IS - 5

ER -