Spectral properties of functional-differential operators and a Gårding-type inequality

Spectral properties of functional-differential operators and a Gårding-type inequality are studied. Studies have shown that if the operator defined for a summation relation satisfies inequality, then, for any nonzero vector, the operator is positive definite. The symmetrized operator is considered which is obtained by extending the coefficients to smooth compactly supported functions. It is shown that the differential operator is strongly elliptic and there exist constants such that the inequality holds for all functions. The functional-differential operator is found to be m-sectorial and associated, Fredholm with index zero, and discrete. Sufficient conditions for operators are obtained to deliver a positive solution Kato's square root problem.