Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients

Let (E, F) be a symmetric non-local Dirichlet from with unbounded coefficient on L-2(R-d; dx) defined by epsilon(f, g) = integral integral(RdxRd) (f (y) - f (x))(g(x) - g(y))W(x, y) J (x, dy) dx, f, g is an element of F, where J(x, dy) is regarded as the jumping kernel for a pure-jump symmetric L ' evy-type process with bounded coefficients, and W(x, y) is seen as a weighted (unbounded) function. We establish sharp criteria for compactness and non-compactness of the associated Markovian semigroup (P-t)(t >= 0) on L-2(R-d; dx). In particular, we prove that if J(x, dy) = | x - y|(-d-alpha) dy with alpha is an element of (0, 2), and W(x, y) = {(1 + |x|)p + (1 + |y|)p, | x - y| < 1 (1 + |x|) q + (1 + |y|) q, | x - y| >= 1 with p is an element of [0, infinity) and q is an element of [0, alpha), then (P-t)(t >= 0) is compact, if and only if p > 2. This indicates that the compactness of (E, F) heavily depends on the growth of the weighted function W(x, y) only for | x - y| < 1. Our approach is based on establishing the essential super Poincar ' e inequality for (E, F). Our general results work even if the jumping kernel J(x, dy) is degenerate or is singular with respect to the Lebesgue measure.