Operator Norm Limits of Order Continuous Operators

Kitover, A. K.; Wickstead, Anthony

doi:10.1007/s11117-005-8513-7

Cited by 11 publications

(14 citation statements)

References 8 publications

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“…If 0 ≤ T ∈ L n (E) and the point r (T ) is a pole of the resolvent R(·, T ) of order m then T −m ∈ L n (E). Indeed, using the identity T −m = lim λ↓r (T ) (λ − r (T )) m R(λ, T ) and the relations 0 ≤ R(λ, T ) ∈ L n (E) for all λ > r (T ), we obtain ( [15]; see also [7]) T −m ∈ L n (E). Thus, every positive order continuous operator T on E is a spectrally order continuous operator.…”

Section: Corollary 27 If An Order Idempotentmentioning

confidence: 99%

The irreducibility in ordered Banach algebras

Alekhno

2011

Positivity

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Let A be an ordered Banach algebra. Put OI(A) = {b ∈where e is a unit of A. An element z ≥ 0 is said to be order continuous if b α ↓ 0 implies b α z ↓ 0 and zb α ↓ 0 for any b α ∈ 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 x y = 0 there exists b ∈ 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 ∈ A is said to be irreducible if for every b ∈ 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 ≤ T ∈ 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).

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Section: Corollary 27 If An Order Idempotentmentioning

confidence: 99%

The irreducibility in ordered Banach algebras

Alekhno

2011

Positivity

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“…Definition 7. 45. Let ((E n , · n ) : n ∈ N) be a multi-normed space, and let K be a closed family of small decompositions of E. Then the multi-norm is orthogonal with respect to K if (x 1 , .…”

Section: Orthogonality With Respect To Familiesmentioning

confidence: 99%

Multi-normed spaces

Dales¹,

Polyakov²

2012

Dissertationes Math.

142

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We modify the very well known theory of normed spaces (E, · ) within functional analysis by considering a sequence ( · n : n ∈ N) of norms, where · n is defined on the product space E n for each n ∈ N.Our theory is analogous to, but distinct from, an existing theory of 'operator spaces'; it is designed to relate to general spaces L p for p ∈ [1, ∞], and in particular to L 1 -spaces, rather than to L 2 -spaces.After recalling in Chapter 1 some results in functional analysis, especially in Banach space, Hilbert space, Banach algebra, and Banach lattice theory, that we shall use, we shall present in Chapter 2 our axiomatic definition of a 'multi-normed space' ((E n , · n ) : n ∈ N), where (E, · ) is a normed space. Several different, equivalent, characterizations of multi-normed spaces are given, some involving the theory of tensor products; key examples of multi-norms are the minimum, maximum, and (p, q)-multi-norms based on a given space. Multi-norms measure 'geometrical features' of normed spaces, in particular by considering their 'rate of growth'. There is a strong connection between multi-normed spaces and the theory of absolutely summing operators.A substantial number of examples of multi-norms will be presented.Following the pattern of standard presentations of the foundations of functional analysis, we consider generalizations to 'multi-topological linear spaces' through 'multi-null sequences', and to 'multi-bounded' linear operators, which are exactly the 'multi-continuous' operators. We define a new Banach space M(E, F ) of multi-bounded operators, and show that it generalizes well-known spaces, especially in the theory of Banach lattices.We conclude with a theory of 'orthogonal decompositions' of a normed space with respect to a multi-norm, and apply this to construct a 'multi-dual' space.Applications of this theory will be presented elsewhere.

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“…Actually, there exists [13] the example of a sequence of order continuous positive operators which converge in norm to a positive operator which is not order continuous (even σ-order continuous). Here the following result holds (see [13], Theorem 2.16, where the given fact was established for rather other class than the class of order continuous operators, while in the our case the proof of it is analogous and will be omitted).…”

Section: The Order Continuity Of the Residuementioning

confidence: 99%

“…The band B of E is called a projection band ([8, p. 32], [16, pp. 61, 135]) whenever B ⊕ B d = E. For a Banach lattice E the Lorenz seminorm (see [13]) on E is defined by the formula…”

Section: Introductionmentioning

confidence: 99%

The Lower Weyl Spectrum of a Positive Operator

Alekhno

2010

Integr. Equ. Oper. Theory

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For the lower Weyl spectrumwhere T is a positive operator on a Banach lattice E, the conditions for which the equality σ − w (T ) = σ − w (T * ) holds, are established. In particular, it is true if E has order continuous norm. An example of a weakly compact positive operator T on ∞ such that the spectral radiusis the Fredholm spectrum, is given. The conditions which guarantee the order continuity of the residue T−1 of the resolvent R(., T ) of an order continuous operator T ≥ 0 at r(T ) / ∈ σ f (T ), are discussed. For example, it is true if T is o-weakly compact. It follows from the proven results that a Banach lattice E admitting an order continuous operator T ≥ 0, r(T ) / ∈ σ f (T ), can not have the trivial band E ∼ n of order continuous functionals in general. It is obtained that a non-zero order continuous operator T : E → F can not be approximated in the r-norm by the operators from E ∼ σ ⊗F , where F is a Banach lattice, E ∼ σ is a disjoint complement of the band E ∼ n of E * .

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Operator Norm Limits of Order Continuous Operators

Cited by 11 publications

References 8 publications

The irreducibility in ordered Banach algebras

The irreducibility in ordered Banach algebras

Multi-normed spaces

The Lower Weyl Spectrum of a Positive Operator

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