The spectral radius of a certain parametric family of functional operators

K e−ihξ dν(h) is the characteristic function of the measure ν. It was also shown that when (2) fails, there can be an infinite-dimensional kernel in this problem. The problem (1) is a natural generalization of boundary-value problems for elliptic differential-difference equations [5], [6] and functional-differential equations with contracted/extended independent variables [2], [3]. We note a connection between (possibly degenerate) elliptic functional-differential operators and Kato’s well-known problem of the square root of a regular accretive operator [6], [7].