SPECTRAL PROPERTIES OF OPERATOR -DIV (-DELTA)-1GRAD IN L2(OMEGA)/R

We are concerned with the spectrum sigma(T) of the operator T = -div(-DELTA)-1 grad in L2 (OMEGA)/R. We prove that sigma(T) subset-or-equal-to [P(OMEGA)-1, 1] where P(OMEGA) is the best constant on the coercivity inequality Absolute value of u 2/L2(OMEGA) less-than-or-equal-to C(OMEGA) parallel-to grad (u) parallel-to 2/H-1(OMEGA))n, for-all u is-an-element-of L2 (OMEGA)/R. P(OMEGA)-1 is an element of the spectrum and parallel-to T parallel-to = 1 is an eigenvalue of T. If OMEGA is the unity ball in R2 or R3, we give the spectrum of the operator T.