C-supercyclic versus R^(+)-supercyclic operators by Bermudez T., Bonilla A., Peris A.

By Bermudez T., Bonilla A., Peris A.

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P* We want to prove that this case can not be happened. Let q^supl-F^I2, then f < p. <9\ P,(J) S U p p,(w\ Pf(z) | ^ | 2 < z+a ^. 5) (t - *)), we get in Pf(z). 6) 2 id = •£&! l to a = f, it yields ^ < £(¥(fl)/flQ + eYM0 + YM0R) < ^ ( * ( i ? ) + eYM0 + YM0R), so 1 < C(9{R) + EYM0 + YM0R) < C(e0 + sYM0). Now we obtain a contradiction for small en and e > 0. (ii) Case II: supl-F^ 2 < l/p2+a. 13). ps < 4. 38 REFERENCES [CS] Y. M. Chen and C. L.

If the scalar curvature is positive, we can prove that the operator D is elliptic. 4) n D(N,a) = - Y, i nH5 ki - hkl)Y(hkihu)(N,a). From Maximum Principle, we obtain that (N, a) is constant. Therefore, (N, a) > 0 or (JV, a) = 0. 2 holds. In particular, when n = 3, the author and Wan [8] proved that the assertion due to Nomizu and Smyth [26] is true without the assumption on the sectional curvature. 3 (Cheng and Wan [8]). Let Mn be an oriented complete hypersurface in E 4 with constant scalar curvature.

Chen Zhihua and Lu Zhiqin M proved: Let M be an n-dimensional complete noncompact Riemannian manifold with a pole and nonnegative radial Ricci curvature outside a compact set, then its essential spectrum of the Laplacian cr es5 (A) = [0, +oo) But we find no result on the essential spectrum of the Laplacian under the assumption that Ricci curvature outside a compact set is not nonnegative. In this paper, we have the following Theorem. Theorem 1. Let M be an n-dimensional complete noncompact Riemannian manifold with a pole P, and outside a compact set K, the radial Ricci curvature > — 4 / " Z Q ^ > where K C B(P,a),r denotes the distance to P, then aess(A) = [0,+oo).

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