Grenzordnungen von Operatorenidealen (II).
Generalizing A. Grothendieck’s (1955) and V. B. Lidskiĭ’s (1959) trace formulas, we have shown in a recent paper that for p ∈ [1,∞] and s ∈ (0,1] with 1/s = 1 + |1/2-1/p| and for every s-nuclear operator T in every subspace of any -space the trace of T is well defined and equals the sum of all eigenvalues of T. Now, we obtain the analogous results for subspaces of quotients (equivalently: for quotients of subspaces) of -spaces.
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 .
We characterise the boundedness of the calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if generates a bounded analytic semigroup on a UMD space, then the calculus of is bounded if and only if has a dilation to a bounded group on . This generalises a Hilbert space result of C.LeMerdy. If is an space we can choose another space in place of .
Let be the realization () of a differential operator on with general boundary conditions (). Here is a homogeneous polynomial of order in complex variables that satisfies a suitable ellipticity condition, and for is a homogeneous polynomial of order...
We give a concise exposition of the basic theory of functional calculus for N-tuples of sectorial or bisectorial operators, with respect to operator-valued functions; moreover we restate and prove in our setting a result of N. Kalton and L. Weis about the boundedness of the operator when f is an R-bounded operator-valued holomorphic function.
Let A be a linear closed densely defined operator in a complex Banach space X. If A is of type ω (i.e. the spectrum of A is contained in a sector of angle 2ω, symmetric around the real positive axis, and is bounded outside every larger sector) and has a bounded inverse, then A has a bounded functional calculus in the real interpolation spaces between X and the domain of the operator itself.
Let A be a linear closed one-to-one operator in a complex Banach space X, having dense domain and dense range. If A is of type ω (i.e.the spectrum of A is contained in a sector of angle 2ω, symmetric about the real positive axis, and is bounded outside every larger sector), then A has a bounded functional calculus in the real interpolation spaces between X and the intersection of the domain and the range of the operator itself.