Extensions of an AC(σ)AC(\sigma) functional calculus

On a reflexive Banach space $X$, if an operator $T$ admits a functional calculus for the absolutely continuous functions on its spectrum $\sigma(T) \subseteq \mathbb{R}$, then this functional calculus can always be extended to include all the functions of bounded variation. This need no longer be true on nonreflexive spaces. In this paper, it is shown that on most classical separable nonreflexive spaces, one can construct an example where such an extension is impossible. Sufficient conditions are also given which ensure that an extension of an \AC functional calculus is possible for operators acting on families of interpolation spaces such as the $L^p$ spaces

A note on positive AN\mathcal{AN} operators

We show that positive absolutely norm attaining operators can be characterized by a simple property of their spectra. This result clarifies and simplifies a result of Ramesh. As an application we characterize weighted shift operators which are absolutely norm attaining.Comment: 2 page

Compact AC(σ)AC(\sigma) operators

All compact $AC(\sigma)$ operators have a representation analogous to that for compact normal operators. As a partial converse we obtain conditions which allow one to construct a large number of such operators. Using the results in the paper, we answer a number of questions about the decomposition of a compact $AC(\sigma)$ into real and imaginary parts

Operational calculus and integral transforms for groups with finite propagation speed

Let $A$ be the generator of a strongly continuous cosine family $(\cos (tA))_{t\in {\bf R}}$ on a complex Banach space $E$. The paper develops an operational calculus for integral transforms and functions of $A$ using the generalized harmonic analysis associated to certain hypergroups. It is shown that characters of hypergroups which have Laplace representations give rise to bounded operators on $E$. Examples include the Mellin transform and the Mehler--Fock transform. The paper uses functional calculus for the cosine family $\cos( t\sqrt {\Delta})$ which is associated with waves that travel at unit speed. The main results include an operational calculus theorem for Sturm--Liouville hypergroups with Laplace representation as well as analogues to the Kunze--Stein phenomenon in the hypergroup convolution setting.Comment: arXiv admin note: substantial text overlap with arXiv:1304.5868. Substantial revision to version

Compact well-bounded operators

Every compact well-bounded operator has a representation as a linear combination of disjoint projections reminiscent of the representation of compact self-adjoint operators. In this note we show that the converse of this result holds, thus characterizing compact well-bounded operators. We also apply this result to study compact well-bounded operators on some special classes of Banach spaces such as hereditarily indecomposable spaces and certain spaces constructed by G. Pisier

A model of suspense for narrative generation

Most work on automatic generation of narratives, and more specifically suspenseful narrative, has focused on detailed domain-specific modelling of character psychology and plot structure. Recent work on the automatic learning of narrative schemas suggests an alternative approach that exploits such schemas for modelling and measuring suspense. We propose a domain-independent model for tracking suspense in a story which can be used to predict the audience’s suspense response on a sentence-by-sentence basis at the content determination stage of narrative generation. The model lends itself as the theoretical foundation for a suspense module that is compatible with alternative narrative generation theories. The proposal is evaluated by human judges’ normalised average scores correlate strongly with predicted values

Approximation in AC(σ)AC(\sigma)

In order to extend the theory of well-bounded operators to include operators with nonreal spectrum, Ashton and Doust introduced definitions for two new algebras of functions defined on a nonempty compact subset $\sigma$ of the plane. These are the functions of bounded variation and the absolutely continuous functions on $\sigma$. Proofs involving absolutely continuous functions usually require that one first works with elements of a dense subset and then take limits. In this paper we present some new theorems about approximating absolutely continuous functions as well as providing missing proofs for some important earlier results.Comment: 23 page

Isomorphisms of AC(σ)AC(\sigma) spaces for linear graphs

We show that among compact subsets of the plane which are drawings of linear graphs, two sets $\sigma$ and $\tau$ are homeomorphic if and only if the corresponding spaces of absolutely continuous functions (in the sense of Ashton and Doust) are isomorphic as Banach algebras. This gives an analogue for this class of sets to the well-known result of Gelfand and Kolmogorov for $C(\Omega)$ spaces.Comment: 18 pages. Minor revisions. To appear in Adv. Oper, Theor