This book gives a gentle but up-to-date introduction into the theory of operator semigroups (or linear dynamical systems), which can be used with great success to describe the dynamics of complicated phenomena arising in many applications. Positivity is a property which naturally appears in physical, chemical, biological or economic processes. It adds a beautiful and far reaching mathematical structure to the dynamical systems and operators describing these processes. In the first part, the finite dimensional theory in a coordinate-free way is developed, which is difficult to find in literature. This is a good opportunity to present the main ideas of the Perron-Frobenius theory in a way which can be used in the infinite dimensional situation. Applications to graph matrices, age structured population models and economic models are discussed. The infinite dimensional theory of positive operator semigroups with their spectral and asymptotic theory is developed in the second part. Recent applications illustrate the theory, like population equations, neutron transport theory, delay equations or flows in networks. Each chapter is accompanied by a large set of exercises, many of them with solutions. An up-to-date bibliography and a detailed subject index help the interested reader. The book is intended primarily for graduate and master students. The finite dimensional part, however, can be followed by an advanced bachelor with a solid knowledge of linear algebra and calculus.
COBISS.SI-ID: 17812569
For any measurable set $E$ of a measure space $(X, \mu)$, let $P_E$ be the (orthogonal) projection on the Hilbert space $L^2(X, \mu)$ with the range $\rm{ran} \, P_E = \{f \in L^2(X, \mu) : f = 0 \ \ a.e. \ on \ E^c\}$ that is called a standard subspace of $L^2(X, \mu)$. Let $T$ be an operator on $L^2(X, \mu)$ having increasing spectrum relative to standard compressions, that is, for any measurable sets $E$ and $F$ with $E \subseteq F$, the spectrum of the operator $P_E T|_{\rm{ran} \, P_E}$ is contained in the spectrum of the operator $P_F T|_{\rm{ran} \, P_F}$. In 2009, Marcoux, Mastnak and Radjavi asked whether the operator $T$ has a non-trivial invariant standard subspace. They answered this question affirmatively when either the measure space $(X, \mu)$ is discrete or the operator $T$ has finite rank. We study this problem in the case of trace-class kernel operators. We also slightly strengthen the above-mentioned result for finite-rank operators.
COBISS.SI-ID: 17797721
Let $\mathcal{S}$ be a semigroup of partial isometries acting on a complex, infinite-dimensional, separable Hilbert space. In this paper we seek criteria which will guarantee that the selfadjoint semigroup $\mathcal{T}$ generated by $\mathcal{S}$ consists of partial isometries as well. Amongst other things, we show that this is the case when the set $\mathcal{Q}(\mathcal{S})$ of final projections of elements of $\mathcal{S}$ generates an abelian von Neumann algebra of uniform finite multiplicity.
COBISS.SI-ID: 17801049
Ever since the historical Sklar's theorem in 1959 copulas have been one of the main tools in modelling dependence of random variables. With the range of applications in applied mathematics expanding and varying from mathematics of finance through system theory to fuzzy set theory, there is a growing need for new types of copulas that could serve as appropriate models in these applications. It is our aim to turn a new page in constructing copulas by setting a counterpart to the famous Marshall copulas (an extension of Marshall-Olkin copulas) that are typically applied to model lifetime of a two-component system. Even a small but essential change of assumptions that the model is applied to such a system with one of the components having a recovery option turns into a substantially different problem on the level of copulas. We give a full study of the augmented case by introducing a new type of copulas, called maxmin, together with concrete applications in server-side web framework, in an economic model and in fuzzy set theory.
COBISS.SI-ID: 17160793
It is shown that the commuting graph of a matrix algebra over a finite field has diameter at most five if the size of the matrices is not a prime nor a square of a prime. It is further shown that the commuting graph of even-sized matrices over finite field has diameter exactly four. This partially proves a conjecture stated by Akbari, Mohammadian, Radjavi, and Raja [Linear Algebra Appl. 418 (2006) 161-176].
COBISS.SI-ID: 17445465