done if we show that 02N/cOT 2 # 0 for T = T~k and E = vk(T~). If this were not the case, then for the same T and E we would obtain the equation V'(T)(Ul A u~ --u~ A u2) = 0 by a straightforward computation, which contradicts (I).Now it is easy to complete the proof of the theorem. First, let 0 < T < T~k+l. Then, by virtue of (III), V(T) < V(T~+I ) = vk+l(T~k+l ) < vk+l(T), and hence, the lemma implies #k(T) < Vk+l(T). By passing to the limit, we obtain #k(T~k+l ) < I/k+l(T~k+l ) . Now let T > T~+I. Then, by virtue of (II) and (III), Vk+l(T) > vk+,(T~k+l ) > pk(T~+l) > p~(T). It remains to check that pk(T~k+l ) ~ Vk+l(T~+l ) . If this is not the case, then it readily follows from the equations pk(T~k+l ) ----vk+l(T~k+l ) and v[+l(T~+l ) = 0 and from the inequalities proved above that p~(T~k+l ) ----0, but this contradicts (II). The theorem is thereby proved. O Remark. Since the potential U is even, we actually have pk -)~k+l and vk+l = ~k+l. It was proved in [4] that in the subspace E0 C L2(-T, T) of 2T-periodic even potentials U with zero mean on the interval [-T, T], the correspondence U ~-+ (vk+l -Pk)k>l is a real-analytic isomorphism of E0 onto l 2. Our results provide examples of potentials whose image under this isomorphism is contained in the positive cone l~.. We must also point out that, unlike [3, 4], our results do not allow us to obtain a potential from a prescribed set of spectral data.Let us finally remark that the rigorous justification of results concerning the absence of the edge state backscattering in the quantum Hall effect [6] is reduced to a proof of the inequality pj,(T) < vj,+~(T).These inequalities were obtained by numerical methods in a series of papers.In conclusion, the authors are grateful to the referee for his suggestions, which made for an improvement of our manuscript.South-Ukrainian Pedagogical University; Odessa State Academy of Civil Engineering and Architecture.
Abstract. Let A and A 0 be linear continuously invertible operators on a Hilbert space H such that A −1 − A −1 0 has finite rank. Assuming that σ(A 0 ) = ∅ and that the operator semigroup V + (t) = exp{iA 0 t}, t 0, is of class C 0 , we state criteria under which the semigroups U ± (t) = exp{±iAt}, t 0, are of class C 0 as well. The analysis in the paper is based on functional models for nonself-adjoint operators and techniques of matrix Muckenhoupt weights.Key words: nonself-adjoint operator, perturbation of a semigroup, functional model, Muckenhoupt condition.
To the centenary of M. G. KreinIn this paper, we apply functional models of nonself-adjoint operators and the technique of matrix Muckenhoupt weights to the theory of one-parameter operator semigroups in Hilbert spaces.1. w-perturbations of linear operators. Let A 0 and A be linear unbounded densely defined and continuously invertible operators on a Hilbert space H such thatwhere f k , g k ∈ H, 1 k n, and the parentheses stand for the inner product in H. We shall assume that the operator A 0 belongs to the class Σ (exp) , that is, and {g k } n 1 such that the operator A related to A 0 ∈ Σ (exp) by (1) generates a C 0 -semigroup U + (t) := exp{iAt} or a C 0 -semigroup U − (t) := exp{−iAt}, t 0. In this paper, we indicate a class of finite-dimensional perturbations A satisfying (1) in which this problem admits a solution.An n × n matrix weight almost everywhere positive on the real line will be denoted by w 2 (x), x ∈ R. Further, by M 2 n we denote the class of matrix weights w 2 satisfying Muckenhoupt's conditionwhere M (w ±2 ) := |∆| −1 ∆ w ±2 (x) dx, ∆ is an arbitrary interval in R, and |∆| is its length. Using the operator A 0 ∈ Σ (exp) and the vectors {g k } n 1 , we construct the n-dimensional rowand introduce the following definition. Definition 1. Suppose that the operators A and A 0 are related by (1).We say that A is a w-perturbation of rank n of the operator A 0 ∈ Σ (exp) if there exists a weight w 2 (x) of the class M 2 n such that the following conditions hold: 1. For each h ∈ H, the function A 0 (x, h)w −1 (x) belongs to L 2 (R).
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