Inverse-Definiteness of the Fourth-Order Symmetric Differential Operator (I)

We give a linear symmetric differential operator L defined by L:=D4*bD2*aI in the 2-periodic function space, and study the inverse-definiteness property of L. We obtain a complete result about the inverse-definiteness property of L with real constants a and b when b 2-4a > 0 and a - bk 2 * k 4 0 for any k {1, 2, 3, . . . }.