Grothendieck's theorem and factorization for operators in Jordan triples.
We deal with a class on nonlinear Schrödinger equations (NLS) with potentials , , and , . Working in weighted Sobolev spaces, the existence of ground states belonging to is proved under the assumption that for some . Furthermore, it is shown that are spikes concentrating at a minimum point of , where .
We investigate the conjugate indicator diagram or, equivalently, the indicator function of (frequently) hypercyclic functions of exponential type for differential operators. This gives insights into growth conditions for these functions on particular rays or sectors. Our research extends known results in several respects.
We show that the growth rates of solutions of the abstract differential equations ẋ(t) = Ax(t), , and the difference equation are closely related. Assuming that A generates an exponentially stable semigroup, we show that on a general Banach space the lowest growth rate of the semigroup is O(∜t), and for it is O(∜n). The similarity in growth holds for all Banach spaces. In particular, for Hilbert spaces the best estimates are O(log(t)) and O(log(n)), respectively. Furthermore, we give conditions...
It will be proved that if is a bounded nilpotent operator on a Banach space of order , where is an integer, then the -th order Cesàro mean and Abel mean of the uniformly continuous semigroup of bounded linear operators on generated by , where , satisfy that (a) for all ; (b) for all ; (c) . A similar result will be also proved for the uniformly continuous cosine function of bounded linear operators on generated by .