Vector valued absolutely continuous functions on idempotent semigroups

AbstractIn this paper we give a description of separating or disjointness preserving linear bijections on spaces of vector-valued absolutely continuous functions defined on compact subsets of the real line. We obtain that they are continuous and biseparating in the finite-dimensional case. The infinite-dimensional case is also studied.

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Absolute Continuity of Some Vector Functions and Measures

Canadian Journal of Mathematics ◽

10.4153/cjm-1972-070-5 ◽

1972 ◽

Vol 24 (5) ◽

pp. 737-746 ◽

Cited By ~ 3

Author(s):

William J. Knight

Keyword(s):

Bounded Variation ◽

Uniform Boundedness ◽

Continuous Functions ◽

Absolutely Continuous Functions ◽

Absolutely Continuous ◽

Continuous Point ◽

Uniform Boundedness Principle ◽

Vector Valued Functions ◽

Analogous Theorem ◽

Vector Valued

In the theory of vector valued functions there is a theorem which states that if a function from a compact interval I into a normed linear space X is of weak bounded variation, then it is of bounded variation. The proof uses in a straightforward way the Uniform Boundedness Principle (see [2, p. 60]). The present paper grew from the question of whether an analogous theorem holds for absolutely continuous functions. The answer is in the negative, and an example will be given (Theorem 7). But it will also be shown that if X is weakly sequentially complete (e.g. an Lp space, 1 ≦ p < ∞ ), then a weakly absolutely continuous point function from / into X is absolutely continuous. The method of proof involves the construction of a countably additive set function in the standard Lebesgue-Stieltjes fashion.The paper is divided into three parts. In Section 1 extensions of finitely additive, absolutely continuous set functions are carried out in an abstract setting. Section 2 applies this to vector valued (point) functions on the real line.

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Vector Valued Absolutely Continuous Functions on Idempotent Semigroups

Transactions of the American Mathematical Society ◽

10.2307/1996365 ◽

1972 ◽

Vol 172 ◽

pp. 491

Author(s):

Richard A. Alo ◽

Andre De Korvin ◽

Richard J. Easton

Keyword(s):

Continuous Functions ◽

Absolutely Continuous Functions ◽

Absolutely Continuous ◽

Vector Valued

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On Finite Part Integrals and Hadamard-Type Fractional Derivatives

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4037930 ◽

2018 ◽

Vol 13 (9) ◽

Cited By ~ 8

Author(s):

Li Ma ◽

Changpin Li

Keyword(s):

Fractional Derivative ◽

Singular Integral ◽

Fractional Derivatives ◽

Continuous Functions ◽

Finite Part ◽

Absolutely Continuous Functions ◽

Absolutely Continuous ◽

Fundamental Properties ◽

Logarithmic Series ◽

The Right

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

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The spectral theorem for well-bounded operators

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700031827 ◽

1993 ◽

Vol 54 (3) ◽

pp. 334-351 ◽

Cited By ~ 1

Author(s):

Ian Doust ◽

Qiu Bozhou

Keyword(s):

Functional Calculus ◽

Continuous Functions ◽

Weak Compactness ◽

Compact Interval ◽

Spectral Theorem ◽

Type B ◽

Bounded Operators ◽

Absolutely Continuous Functions ◽

Absolutely Continuous

AbstractWell-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.

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