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On spectral continuity of positive elements

100%

Mouton S.

Studia Mathematica

2006

tom 174

nr 1

75-84

Let x be a positive element of an ordered Banach algebra. We prove a relationship between the spectra of x and of certain positive elements y for which either xy ≤ yx or yx ≤ xy. Furthermore, we show that the spectral radius is continuous at x, considered as an element of the set of all positive elements y ≥ x such that either xy ≤ yx or yx ≤ xy. We also show that the property ϱ(x + y) ≤ ϱ(x) + ϱ(y) of the spectral radius ϱ can be obtained for positive elements y which satisfy at least one of the above inequalities.

Domination properties in ordered Banach algebras

63%

Mouton H., Mouton S.

Studia Mathematica

2002

tom 149

nr 1

63-73

We recall from [9] the definition and properties of an algebra cone C of a real or complex Banach algebra A. It can be shown that C induces on A an ordering which is compatible with the algebraic structure of A. The Banach algebra A is then called an ordered Banach algebra. An important property that the algebra cone C may have is that of normality. If C is normal, then the order structure and the topology of A are reconciled in a certain way. Ordered Banach algebras have interesting spectral properties. If A is an ordered Banach algebra with a normal algebra cone C, then an important problem is that of providing conditions under which certain spectral properties of a positive element b will be inherited by positive elements dominated by b. We are particularly interested in the property of b being an element of the radical of A. Some interesting answers can be obtained by the use of subharmonic analysis and Cartan's theorem.

Ograniczanie wyników

2 Studia Mathematica

2 Mouton S.

1 Mouton H.

1 2006

1 2002

Wyniki wyszukiwania

On spectral continuity of positive elements

Domination properties in ordered Banach algebras