On an Estimate of the Resolvent of an Even-Order Operator Bundle and its Application

This paper investigates an estimate of the resolvent for a class of even-order operator bundles acting in a scale of Hilbert spaces generated by a positive definite self-adjoint operator. The operator bundle has coefficients that exhibit symmetry with respect to the parity of their indices. Additionally, a connection is established between the spectral properties of the operator bundle and the existence of solutions to abstract differential equations. This provides practical applicability for the obtained theorems in stability analysis, spectral computations, and the construction of numerical schemes for parameter-dependent problems. The proposed methods and results extend the classical approaches of Lions, Vishik, and Agranovich to high-dimensional and generalized operator systems. It is demonstrated that, even in the absence of compactness, precise resolvent estimates can be obtained, and spectral purity along the imaginary axis can be ensured.