This paper considers three dichotomy concepts (exponential dichotomy, uniform
exponential dichotomy and strong exponential dichotomy) in the general context
of non-invertible evolution operators in Banach spaces. Connections between
these concepts are illustrated. Using the notion of Green function, we give
necessary conditions and sufficient ones for strong exponential dichotomy. Some
illustrative examples are presented to prove that the converse of some
implication type theorems are not valid
We study the problem of whether all bipartite quantum states having a
prescribed spectrum remain positive under the reduction map applied to one
subsystem. We provide necessary and sufficient conditions, in the form of a
family of linear inequalities, which the spectrum has to verify. Our conditions
become explicit when one of the two subsystems is a qubit, as well as for
further sets of states. Finally, we introduce a family of simple entanglement
criteria for spectra, closely related to the reduction and positive partial
transpose criteria, which also provide new insight into the set of spectra that
guarantee separability or positivity of the partial transpose.Comment: Linear Algebra and its Applications (2015
Abstract. In this paper we study the reduction criterion for detecting entanglement of large dimensional bipartite quantum systems. We first obtain an explicit formula for the moments of a random quantum state to which the reduction criterion has been applied. We show that the empirical eigenvalue distribution of this random matrix converges strongly to a limit that we compute, in three different asymptotic regimes. We then employ tools from free probability theory to study the asymptotic positivity of the reduction operators. Finally, we compare the reduction criterion with other entanglement criteria, via thresholds.
For a discrete dynamics defined by a sequence of bounded and not necessarily invertible linear operators, we give a complete characterization of exponential stability in terms of invertibility of a certain operator acting on suitable Banach sequence spaces. We connect the invertibility of this operator to the existence of a particular type of admissible exponents. For the bounded orbits, exponential stability results from a spectral property. Some adequate examples are presented to emphasize some significant qualitative differences between uniform and nonuniform behavior.2010 Mathematics Subject Classification. 39A30, 37D25.
This paper considers two trichotomy concepts in the context of abstract evolution operators. The first one extends the notion of exponential trichotomy in the sense of Elaydi-Hajek for differential equations to abstract evolution operators, and it is a natural extension of the generalized exponential dichotomy considered in the paper by Jiang (2006). The second concept is dual in a certain sense to the first one. We prove that these types of exponential trichotomy imply the existence of generalized exponential dichotomy on both half-lines. We emphasize that we do not assume the invertibility of the evolution operators on the whole spaceX(unlike the case of evolution operators generated by differential equations).
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