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Functions of bounded variation on compact subsets of the plane

100%
Ashton B., Doust I.
Studia Mathematica
|
2005
|
tom 169
|
nr 2
163-188
EN
A major obstacle in extending the theory of well-bounded operators to cover operators whose spectrum is not necessarily real has been the lack of a suitable variation norm applicable to functions defined on an arbitrary nonempty compact subset σ of the plane. In this paper we define a new Banach algebra BV(σ) of functions of bounded variation on such a set and show that the function-theoretic properties of this algebra make it better suited to applications in spectral theory than those used previously.
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Compact AC-operators

100%
Doust I., Walden B. L.
Studia Mathematica
|
1995-1996
|
tom 117
|
nr 3
275-287
EN
We prove that compact AC-operators have a representation as a combination of disjoint projections which mirrors that for compact normal operators. We also show that unlike arbitrary AC-operators, compact AC-operators admit a unique splitting into real and imaginary parts, and that these parts must necessarily be compact.
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Isomorphisms of AC(σ) spaces

100%
Doust I., Leinert M.
Studia Mathematica
|
2015
|
tom 228
|
nr 1
7-31
EN
Analogues of the classical Banach-Stone theorem for spaces of continuous functions are studied in the context of the spaces of absolutely continuous functions introduced by Ashton and Doust. We show that if AC(σ₁) is algebra isomorphic to AC(σ₂) then σ₁ is homeomorphic to σ₂. The converse however is false. In a positive direction we show that the converse implication does hold if the sets σ₁ and σ₂ are confined to a restricted collection of compact sets, such as the set of all simple polygons.
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