We consider conditions under which a continuous functional calculus for a Banach space operator T ?? L(X) may be extended to a bounded Borel functional calculus, and under which a functional calculus for absolutely continuous (AC) functions may be extended to one of for functions of bounded variation (BV). The natural setting for investigating the former case is finitely spectral operators, and for the latter, well-bounded operators. Some such conditions are well-established. If X is a reflexive space, both type of Extensions are assured; in fact if X contains an isomorphic copy of co, then every Operator T ?? L(X) that has a continuous functional calculus necessarily admits a Borel one. We show that if a space X has a predual, then also every operator T ?? L(X) with a continuous functional calculus admits a bounded Borel functional Calculus. In case a Banach space X either contains an isomorphic copy of co, or has a Predual, and T ?? L(X) is an operator with an AC functional calculus, we find that the existence of a decomposition of the identity of bounded variation for T is sufficient to ensure that the AC functional calculus may be extended to a BV functional calculus. We also consider operators defined by a linear map on interpolation families of Banach spaces [Xr, X???] (r???1), where for example Xp = lp, Lp[0,1] or Cp. We show that under certain uniform boundedness conditions, the possession of a BV functional calculus by operators on the spaces Xp, p ?? (r, ???), may be extrapolated to the corresponding operators on the spaces Xr and X???.

Thesis (Ph. D.)--Oregon State University, 1968. / Typescript (photocopy). Includes bibliographical references (leaves 60-62). Also available on the World Wide Web.

Thesis (Ph. D.)--University of Missouri-Columbia, 2000. / Typescript. Vita. Includes bibliographical references (leaves 149-155). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2000. / Typescript. Vita. Includes bibliographical references (leaves 149-155). Also available on the Internet.

No description available.

We consider conditions under which a continuous functional calculus for a Banach space operator T ?? L(X) may be extended to a bounded Borel functional calculus, and under which a functional calculus for absolutely continuous (AC) functions may be extended to one of for functions of bounded variation (BV). The natural setting for investigating the former case is finitely spectral operators, and for the latter, well-bounded operators. Some such conditions are well-established. If X is a reflexive space, both type of Extensions are assured; in fact if X contains an isomorphic copy of co, then every Operator T ?? L(X) that has a continuous functional calculus necessarily admits a Borel one. We show that if a space X has a predual, then also every operator T ?? L(X) with a continuous functional calculus admits a bounded Borel functional Calculus. In case a Banach space X either contains an isomorphic copy of co, or has a Predual, and T ?? L(X) is an operator with an AC functional calculus, we find that the existence of a decomposition of the identity of bounded variation for T is sufficient to ensure that the AC functional calculus may be extended to a BV functional calculus. We also consider operators defined by a linear map on interpolation families of Banach spaces [Xr, X???] (r???1), where for example Xp = lp, Lp[0,1] or Cp. We show that under certain uniform boundedness conditions, the possession of a BV functional calculus by operators on the spaces Xp, p ?? (r, ???), may be extrapolated to the corresponding operators on the spaces Xr and X???.

No description available.

A continuous convex function on an open interval of the real line is differentiable everywhere except on a countable subset of its domain. There has been interest in the problem of characterizing those Banach spaces where the continuous functions exhibit similar differentiability properties. In this paper we show that if a Banach space E has property (H*) and B_E• is weak* sequentially compact, then E is an Asplund space. In the case where the space is weakly compactly generated, it is shown that property (H*) is equivalent for the space to admit an equivalent Frechet differentiable norm. Moreover, we define the SH* spaces, show that every SH* space is an Asplund space, and show that every weakly sequentially complete SH* space is reflexive. Also, we study the relation between property (H*) and the asymptotic norming property (ANP). By a slight modification of the ANP we define the ANP*, and show that if the dual of a Banach spaces has the ANP*-I then the space admits an equivalent Fréchet differentiability norm, and that the ANP*-II is equivalent to the space having property (H*) and the closed unit ball of the dual is weak* sequentially compact. Also, we show that in the dual of a weakly countably determined Banach space all the ANP-K'S are equivalent, and they are equivalent for the predual to have property (H*).

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This thesis investigates the properties and applications of derivatives of functions whose domain and range are Banach spaces.