�ber das Spektrum positiver Operatoren

Lotz, Heinrich P.

doi:10.1007/bf01110453

Cited by 75 publications

(28 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…A decomposition of T finally results which generalizes the Frobenius normal form of a reducible matrix. At this juncture, we should point out that our arguments are somewhat similar to those used by H. Lotz [7] in showing that a nonnegative operator whose resolvent has unity as a pole is ((? )-solvable.…”

Section: -(£ I)-mentioning

confidence: 64%

“…So, in (5.14), the range of (T ί+ i) 7 is precisely equal to the range of T +i, for any 7 = 1,..., and we conclude that (5.14) is solvable for a real φ i+ \ residing in φ z+ i(/o). So, for any real number μ, solves (5.14).…”

Section: Suppose That Z Is a Principal ί-Band With σ(Z)mentioning

confidence: 70%

“…The decomposition of a reducible operator T in §11 will be accomplished by using and refining a bit the techniques, employed by H. P. Lotz [7], to show that an operator, with its spectral radius a pole of the resolvent, is (G ! )-solvable [15, p. 326].…”

Section: ) Y:=σexp(-2πi(q-\)/p)f Qmentioning

confidence: 99%

“…He used very crucially results by H. P. Lotz [7,8] refining the assertions of the NiiroSawashima Theorem, whereby the peripheral spectrum of any positive operator consists of poles of its resolvent, once its spectral radius is assumed a pole with finite-dimensional residuum. With E having an order continuous norm, Axmann's results can be summarized as:…”

mentioning

confidence: 99%

See 3 more Smart Citations

On the ideal structure of positive, eventually compact linear operators on Banach lattices

Jang¹,

Victory²

1993

Pacific J. Math.

View full text Add to dashboard Cite

We study the structure of the algebraic eigenspace corresponding to the spectral radius of a nonnegative reducible linear operator T, having a compact iterate and defined on a Banach lattice E with order continuous norm. The Perron-Frobenius theory is generalized by showing that this algebraic eigenspace is spanned by a basis of eigenelements and generalized eigenelements possessing certain positivity features. A combinatorial characterization of both the Riesz index of the spectral radius and the dimension of the algebraic eigenspace is given. These results are made possible by a decomposition of T, in terms of certain closed ideals of E, in a form which directly generalizes the Frobenius normal form of a nonnegative reducible matrix/ I. Introduction. Let £ bea Banach lattice, and T, a positive reducible linear operator mapping E into itself and having a compact iterate. Suppose, in addition, that r(T), the spectral radius of T, is positive. The primary purpose of this research is to ascertain, under what conditions on the Banach lattice E, a decomposition of T is possible, which turns out to be a natural generalization of the Frobenius normal form for reducible matrices. These properties will be seen to generalize those deduced by U. Rothblum [13] for the matrix setting, and those by H. D. Victory, Jr. for integral operators on LPspaces, 1 < p < oo, with the underlying measure being σ-finite [16, 17] on a domain set Ω.We refer the reader to the treatise by H. H. Schaefer [15] for an explanation of the notation used in this work and of the lattice concepts of ideals, order convergence and completeness, bands, projection bands, operator-invariant ideals, and (uniform) mean ergodicity of an operator T. By £(E), we mean the Banach space of bounded linear endomorphisms of E.The underlying Banach lattice E will be assumed equipped with an order continuous norm. In such Banach lattices, every filter that order converges norm converges. Such lattices are characterized by the fact that every closed ideal is a band [15, Theorem 5.14 (Chapter II)]. Since

show abstract

Section: -(£ I)-mentioning

confidence: 64%

Section: Suppose That Z Is a Principal ί-Band With σ(Z)mentioning

confidence: 70%

Section: ) Y:=σexp(-2πi(q-\)/p)f Qmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

On the ideal structure of positive, eventually compact linear operators on Banach lattices

Jang¹,

Victory²

1993

Pacific J. Math.

View full text Add to dashboard Cite

show abstract

“…For further contributions to this topic we refer the reader to Lotz [19 [21, p. 352] where this result is stated in English) and to Zhang [23,Theorem 2.11]. In Section 8 we briefly deal with C 0 -semigroups of Markov operators and we show that one of our examples in Section 6 can be adapted to the C 0 -semigroup case.…”

Section: If T Is (Ws)-bounded Then the Peripheral Spectrum Of T Is Cmentioning

confidence: 99%

On the peripheral spectrum of positive operators

Glück

2015

Positivity

View full text Add to dashboard Cite

Abstract. This paper contributes to the analysis of the peripheral (point) spectrum of positive linear operators on Banach lattices. We show that, under appropriate growth and regularity conditions, the peripheral point spectrum of a positive operator is cyclic and that the corresponding eigenspaces fulfil a certain dimension estimate. A couple of examples demonstrates that some of our theorems are optimal. Our results on the peripheral point spectrum are then used to prove a sufficient condition for the peripheral spectrum of a positive operator to be cyclic; this generalizes theorems of Lotz and Scheffold.

show abstract

A Radon-nikodym-theorem for operators with an application to spectral theory

Wolff

Foundations of Quantum Mechanics and Ordered Linear Spaces

View full text Add to dashboard Cite

�ber das Spektrum positiver Operatoren

Cited by 75 publications

References 17 publications

On the ideal structure of positive, eventually compact linear operators on Banach lattices

On the ideal structure of positive, eventually compact linear operators on Banach lattices

On the peripheral spectrum of positive operators

A Radon-nikodym-theorem for operators with an application to spectral theory

Contact Info

Product

Resources

About