21 research outputs found
A complete characterization of exponential stability for discrete dynamics
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.Comment: The final version will be published in Journal of Difference
Equations and Application
Positive reduction from spectra
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
Generalized evolution semigroups and general dichotomies
We introduce a special class of real semiflows, which is used to define a
general type of evolution semigroups, associated to not necessarily
exponentially bounded evolution families. Giving spectral characterizations of
the corresponding generators, our results directly apply to a wide class of
dichotomies, such as those with time-varying rate of change
Admissibility and general dichotomies for evolution families
For an arbitrary noninvertible evolution family on the half-line and for
in a large class of rate functions, we
consider the notion of a -dichotomy with respect to a family of norms and
characterize it in terms of two admissibility conditions. In particular, our
results are applicable to exponential as well as polynomial dichotomies with
respect to a family of norms. As a nontrivial application of our work, we
establish the robustness of general nonuniform dichotomies