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A linear operator between two ordered vector spaces that carries positive elements to positive elements is known in the literature as a positive operator. As we have mentioned in the preface, the main theme of this book is the study of positive operators. To obtain fruitful and useful results the domains and the ranges of positive operators will be taken to be Riesz spaces (vector lattices). For this reason, in order to make the material as self-sufficient as possible, the fundamental properties of Riesz spaces are discussed as they are needed.

The basic properties of positive operators were investigated in the previous chapter. In this chapter we shall study the lattice behavior of three specific classes of positive operators.

The basic properties of positive operators were investigated in the previous chapter. In this chapter we shall study the lattice behavior of three specific classes of positive operators.

It is well known that operator theory is intrinsically related to the topological structures associated with the spaces upon which the operators act. The theory of positive operators is no exception to this phenomenon. The various topological notions provide an invaluable insight into the properties of operators. This chapter is devoted to the basic topological concepts needed for the study of positive operators. The presentation (although concise) is quite complete. The discussion focuses on locally convex spaces, Banach spaces, and locally solid Riesz spaces.

It is well known that operator theory is intrinsically related to the topological structures associated with the spaces upon which the operators act. The theory of positive operators is no exception to this phenomenon. The various topological notions provide an invaluable insight into the properties of operators. This chapter is devoted to the basic topological concepts needed for the study of positive operators. The presentation (although concise) is quite complete. The discussion focuses on locally convex spaces, Banach spaces, and locally solid Riesz spaces.

A linear operator between two ordered vector spaces that carries positive elements to positive elements is known in the literature as a positive operator. As we have mentioned in the preface, the main theme of this book is the study of positive operators. To obtain fruitful and useful results the domains and the ranges of positive operators will be taken to be Riesz spaces (vector lattices). For this reason, in order to make the material as self-sufficient as possible, the fundamental properties of Riesz spaces are discussed as they are needed.

We investigate a differential game motivated by a problem in mathematical finance. This game displays two interesting features. On the one hand, one of the players, ursuer say, may, and will, use infinitely large controls, i.e., impulses, producing “jumps” in the state variables. Standard optimal trajectories are made of such a jump followed by a “coasting period” where exerts no control. This leads to barriers of a somewhat new type. But because the cost of jumps is only proportional to their amplitude, some singular optimal trajectories arise where uses an intermediary control, nonzero but finite. (In classical impulse control, there is a minimum positive cost to any use of the control, forbidding such a mixed situation.)

On the other hand, the complete solution of the game exhibits a type of singularity, the existence of which had long been conjectured (noticeably by Arik Melikyan in discussions with the first author) but, as far as we know, never shown in actual examples: a two-dimensional focal manifold traversed by noncollinear optimal fields depending on the control used by vader. It is on this manifold that intermediary controls for arise.

Finally, we show that the Isaacs equation of a discrete-time version of the problem provides a discretization scheme that converges to the value function of the differential game. This is done through the investigation of a (degenerate) quasi-variational inequality and its viscosity solution, with the help of an equivalent, but nonimpulsive, differential game—a method of interest per se that we credit to Joshua—to which we apply essentially the classical method of Capuzzo Dolcetta extended to differential games by Pourtallier and Tidball, with some technical adaptations.

It is well known that most classical Banach spaces are, in fact, Banach lattices on which positive operators appear naturally. This chapter is devoted to the study of Banach lattices with special emphasis on Banach lattices with order continuous norms.

A compact operator sends an arbitrary norm bounded sequence to a sequence with a norm convergent subsequence. For this reason, when operators are associated with integral equations, the compact operators are the most desirable. Besides being compact, an operator with some type of compactness is more useful than an arbitrary operator.

A compact operator sends an arbitrary norm bounded sequence to a sequence with a norm convergent subsequence. For this reason, when operators are associated with integral equations, the compact operators are the most desirable. Besides being compact, an operator with some type of compactness is more useful than an arbitrary operator.