Minimality of root vectors of operator functions analytic in an angle

Abstract

We study the minimality of elements $x_{h, j, k}$ of canonical systems of root vectors. These systems correspond to the characteristic numbers $μ_k$ of operator functions $L(λ)$ analytic in an angle; we assume that operators act in a Hilbert space $H$. In particular, we consider the case where $L(λ) = I + T(λ)C^{β} > 0, \;I$ is an identity operator, $C$ is a completely continuous operator, $∥(I- λC)^{−1}∥ ≤ c$ for $|\arg λ| ≥ θ,\; 0 < θ < π$, the operator function $T(λ)$ is analytic, and $T(λ)$ for $|\arg λ| < θ$. It is proved that, in this case, there exists $ρ > 0$ such that the system of vectors $C^v_{x_{h,j,k}}$ is minimal in $ H$ for arbitrary positive $ν < 1+β,$ provided that $¦μ_k¦ > ρ$.