On lattice isomorphisms with positive real spectrum and groups of positive operators

Schaefer, Helmut; Wolff, Manfred; Arendt, Wolfgang

doi:10.1007/bf01174818

Cited by 29 publications

(16 citation statements)

References 7 publications

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“…By studying the spectrum of lattice homomorphisms, [11] also contains some results about groups of positive operators, in particular it is shown in [11,Corollary 3.10] that a uniformly bounded group of positive operators on a Banach lattice is discrete in the norm topology, a result we obtain in the special case of groups which are compact in the strong operator topology on certain sequence spaces, cf. Corollary 5.6 below.…”

Section: Introduction and Overviewmentioning

confidence: 99%

“…This proposition is stated in [11,Proposition 1.4], where a reference to [5] is given for the proof. The development of the theory since the appearance of [5] enables us to give a proof as above.…”

Section: Preliminariesmentioning

confidence: 99%

“…In [11,Corollary 3.10], by studying the spectrum of lattice homomorphisms, this corollary is shown with δ = √ 3 sup{ T : T ∈ G}, for arbitrary uniformly bounded groups of positive operators in complex Banach lattices. Now we will examine invariant structures under these strongly continuous positive representations.…”

Section: Conversely If H ⊂ S(n) Is a Compact Group And Mmentioning

confidence: 99%

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Compact groups of positive operators on Banach lattices

Jeu

Wortel

2014

Indagationes Mathematicae

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Abstract. In this paper we study groups of positive operators on Banach lattices. If a certain factorization property holds for the elements of such a group, the group has a homomorphic image in the isometric positive operators which has the same invariant ideals as the original group. If the group is compact in the strong operator topology, it equals a group of isometric positive operators conjugated by a single central lattice automorphism, provided an additional technical assumption is satisfied, for which we have only examples. We obtain a characterization of positive representations of a group with compact image in the strong operator topology, and use this for normalized symmetric Banach sequence spaces to prove an ordered version of the decomposition theorem for unitary representations of compact groups. Applications concerning spaces of continuous functions are also considered.

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Section: Introduction and Overviewmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Conversely If H ⊂ S(n) Is a Compact Group And Mmentioning

confidence: 99%

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Compact groups of positive operators on Banach lattices

Jeu

Wortel

2014

Indagationes Mathematicae

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“…Theorem 2.1 of [6] in fact tells us that T\His simply r times the identity on H, so is certainly central.…”

Section: Corollarymentioning

confidence: 99%

Spectral properties of compact lattice homomorphisms

Wickstead¹

1982

Proc. Amer. Math. Soc.

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Abstract. Given two nonzero eigenvalues of a lattice homomorphism on a relatively uniformly complete vector lattice, of different moduli and with at least one isolated in the set of all eigenvalues, we show that corresponding eigenvectors must be disjoint. The analogous result for the approximate point spectrum of a lattice homomorphism on a Banach lattice is deduced. We give an infinite spectral decomposition for a lattice homomorphism, on a Banach lattice with order continuous norm, which is compact and has an adjoint which is also a lattice homomorphism. From this we deduce that if it has nonnegative spectrum, then it is the direct sum of a nilpotent lattice homomorphism and one that is central.

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“…Gelfand proved that if the sequence (/? ")"<=£ is bounded then R = kl (see [1,12]). This result is extended in [2] and [14] to contractions R such that (b') l0g + ||/r"||=0(H5) («->+OO).…”

Section: \\T(t)\\=o(t K+1 ) (T-++oo) (B)mentioning

confidence: 99%