Kernel (set theory)

Abstract: Using the theory of linear relations in Pontryagin spaces we extend to the nonpositive case the theory of reproducing kernel spaces associated with contractions in Hilbert spaces.

Abstract: In this paper we prove that if S equals a finite sum of finite productsof Toeplitz operators on the Bergman space of the unit disk, thenS is compact if and only if the Berezin transform of S equals 0 on∂D. This result is new even when S equals a single Toeplitz operator.Our main result can be used to prove, via a unified approach, severalpreviously known results about compact Toeplitz operators, compactHankel operators, and appropriate products of these operators. 1 Introduction Let dA denote Lebesgue area measure on the unit disk D, normalizedso that the measure of D equals 1. The Bergman space L 2a is the Hilbertspace consisting of the analytic functions on D that are also in L 2 (D,dA).For z ∈ D, the Bergman reproducing kernel is the function K z ∈ L 2a suchthatf(z) = hf,K z ifor every f ∈ L 2a . The normalized Bergman reproducing kernel k z is thefunction K z /kK z k 2 . Here, as elsewhere in this paper, the norm k k 2 and theinner product h , i are taken in the space L 2 (D,dA).For S a bounded operator on L

Abstract: Publisher Summary This chapter focuses on the bargaining set, kernel, and nucleolus. The theory of the bargaining set answers a more modest question: How would or should the players share the proceeds, given that a certain coalition structure (c.s.) has formed? From a normative point of view, the reason for asking such a question stems from the need to let the players know what to expect from each coalition structure so that they can then make up their mind about the coalitions they want to join, and in what configuration. The kernel was introduced as an auxiliary solution concept, the main task of which was to illuminate properties of the bargaining set and to compute at least part of this set. Kernel had many interesting mathematical properties that reflected in various ways the structure of the game. Kernel [prekernel] is covariant with respect to strategic equivalence. Both are finite unions of polytopes. It is almost as difficult to compute the kernel as to compute the bargaining set. Being a point in the kernel, the nucleolus point has all the nice properties of the kernel points. Becauseit is a solution concept that does not depend on the names of the players, it preserves all the symmetries of the game. The nucleolus has an advantage over the Shapley value. Because the nucleolus is not empty even if the core is empty, it can be stated that the nucleolus is the location of the latent position of the core. The chapter also discusses the axiomatic foundation of the prekernel and the prenucleolus, ideas embodied in the bargaining set, kernel, and nucleolus that spawned many other related solution concepts, and dynamic processes that lead the participants in a cooperative game to reach the bargaining set— or the kernel, or the nucleolus, or many other bargaining sets— via a sequence of steps that make good intuitive sense.

Abstract: The decomposition of tensor products of representations into irreducibles is studied for a continuous family of integrable operator representations of $U_q(sl(2,R)$. It is described by an explicit integral transformation involving a distributional kernel that can be seen as an analogue of the Clebsch-Gordan coefficients. Moreover, we also study the relation between two canonical decompositions of triple tensor products into irreducibles. It can be represented by an integral transformation with a kernel that generalizes the Racah-Wigner coefficients. This kernel is explicitly calculated.