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Y. Abramovich, G. Sirotkin (2005)
On Order Convergence of NetsPositivity, 9
C. Aliprantis, O. Burkinshaw (2006)
Positive Operators
M. Krasnoselʹskii, E. Lifshit︠s︡, A. Sobolev (1989)
Positive Linear Systems, the Method of Positive OperatorsActa Applicandae Mathematica, 24
E. Alekhno (2007)
Some properties of essential spectra of a positive operator, IIPositivity, 13
A. Kitover, A. Wickstead (2005)
Operator Norm Limits of Order Continuous OperatorsPositivity, 9
J. Grobler, C. Reinecke (1997)
On principal T-bands in a Banach latticeIntegral Equations and Operator Theory, 28
E. Alekhno (2010)
The Lower Weyl Spectrum of a Positive OperatorIntegral Equations and Operator Theory, 67
W. Luxemburg, A. Zaanem (2012)
ON THE IDEAL STRUCTURE OF POSITIVE , EVENTUALLY COMPACT LINEAR OPERATORS ON BANACH LATTICES
A. Gaur, Mursaleen (1997)
Cones in Banach AlgebrasKyungpook Mathematical Journal, 37
E. Alekhno (2007)
Some Properties of Essential Spectra of a Positive OperatorPositivity, 11
S. Mouton, H. Raubenheimer (1997)
More Spectral Theory in Ordered Banach AlgebrasPositivity, 1
J. Grobler (1995)
Spectral Theory in Banach LatticesOperator theory, 75
V. Caselles (1987)
On the peripheral spectrum of positive operatorsIsr. J. Math., 58
E. Alekhno (2006)
Spectral Properties of Band Irreducible Operators
P. Aiena (2004)
Fredholm and Local Spectral Theory, with Applications to Multipliers
H. Raubenheimer, S. Rode (1996)
Cones in Banach algebrasIndag. Math. N.S., 7
Y. Abramovich, C. Aliprantis (2002)
An invitation to operator theory
I. Burger, J. Grobler (1995)
SPECTRAL PROPERTIES OF POSITIVE ELEMENTS IN BANACH LATTICE ALGEBRASQuaestiones Mathematicae, 18
C. Aliprantis, R. Tourky (2007)
Cones and duality
Let A be an ordered Banach algebra. Put $$\mathbf{OI}(A)=\{b\in A: 0 \le b\le e, b^2 = b\},$$ where e is a unit of A. An element z ≥ 0 is said to be order continuous if $${b_\alpha\downarrow 0}$$ implies $${b_\alpha z \downarrow 0}$$ and $${zb_\alpha\downarrow 0}$$ for any $${b_\alpha \in \mathbf{OI}(A)}$$ . It is shown that if E is a Dedekind complete Banach lattice then the set of all order continuous elements in L(E) coincides with the set of all positive order continuous operators on E. An algebra A is said to have a (strongly) disjunctive product if for any order continuous x and y in A(x, y ≥ 0) with xy = 0 there exists $${b \in \mathbf{OI}(A)}$$ such that xb = (e − b)y = 0. We show that the algebra L(E) has the strongly disjunctive product iff E has order continuous norm. An element $${z\in A}$$ is said to be irreducible if for every $${b \in \mathbf{OI}(A)}$$ the relation (e − b)zb = 0 implies either b = 0 or b = e. We investigate spectral properties of irreducible elements in algebras with a disjunctive product. The spectral radius r(z) is called an f-pole of the resolvent R(·, z) if 0 ≤ x ≤ z implies r(x) ≤ r(z) and if r(x) = r(z) then r(z) is a pole of R(·, x). We show that under some natural assumptions on the Banach lattice E, if $${0\le T \in L(E)}$$ then r(T) is an f-pole of R(·,T) iff r(T) is a finite-rank pole of R(·, T). We also present a theorem about the Frobenius normal form of z when r(z) is an f-pole of R(·, z). Some applications to the spectral theory of irreducible operators and the general spectral theory of positive elements are provided. In particular, we show that under some conditions 0 ≤ x < z implies r(x) < r(z).
Positivity – Springer Journals
Published: Mar 16, 2011
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