Linear Operators in Complete Positive Cones

Bonsall, F. F.

doi:10.1112/plms/s3-8.1.53

Cited by 86 publications

(54 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Generalizations, particularly allowing noncompact T , can be found in [1,9,18,19,21]. Our approach in this paper will be to use generalizations of the Krein-Rutman theorem.…”

Section: Introductionmentioning

confidence: 99%

Positive operators and Hausdorff dimension of invariant sets

Nussbaum

Priyadarshi

Lunel

2012

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. In this paper we obtain theorems which give the Hausdorff dimension of the invariant set for a finite family of contraction mappings which are "infinitesimal similitudes" on a complete, perfect metric space. Our work generalizes the graph-directed construction of Mauldin and Williams (1988) and is related in its general setting to results of Schief (1996), but differs crucially in that the mappings need not be similitudes. We use the theory of positive linear operators and generalizations of the Krein-Rutman theorem to characterize the Hausdorff dimension as the unique value of σ > 0 for which r(L σ ) = 1, where L σ , σ > 0, is a naturally associated family of positive linear operators and r(L σ ) denotes the spectral radius of L σ . We also indicate how these results can be generalized to countable families of infinitesimal similitudes. The intent here is foundational: to derive a basic formula in its proper generality and to emphasize the utility of the theory of positive linear operators in this setting. Later work will explore the usefulness of the basic theorem and its functional analytic setting in studying questions about Hausdorff dimension.

show abstract

“…Generalizations, particularly allowing noncompact T , can be found in [1,9,18,19,21]. Our approach in this paper will be to use generalizations of the Krein-Rutman theorem.…”

Section: Introductionmentioning

confidence: 99%

Positive operators and Hausdorff dimension of invariant sets

Nussbaum

Priyadarshi

Lunel

2012

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Hence if (1/λ) ∈ ρ(U ) also, then Finally, we offer the following proof of a known theorem [4], [8], using approximation by polynomials with nonnegative coefficients. Our belief is that this proof is somewhat more transparent than previous arguments.…”

Section: Is Invertible In Particular If a Is Equipped With A Conjugmentioning

confidence: 97%

“…It is a classical result of Bonsall [4] and Schaefer [8] that if K is normal and generating, then r(A) ∈ σ(A). However, Bonsall [4] gave an example of a closed, generating cone K in a Hilbert space H and a positive bounded linear map A : H → H for which r(A) / ∈ σ(A).…”

Section: Principal Question: Under What Further Conditions On a Is Itmentioning

confidence: 99%

“…However, Bonsall [4] gave an example of a closed, generating cone K in a Hilbert space H and a positive bounded linear map A : H → H for which r(A) / ∈ σ(A). In an extremely interesting recent paper, Toland [12] was able to show that if K is a closed, total cone in a real Hilbert space H and A is a bounded self-adjoint operator on H with A[K] ⊆ K, then r(A) ∈ σ(A) must hold.…”

Section: Principal Question: Under What Further Conditions On a Is Itmentioning

confidence: 99%

“…If K is a closed, total cone in a real Hilbert space H and A is a bounded normal operator on H such that A[K] ⊆ K and σ(i(A − A * )) has 1-dimensional Lebesgue measure zero, then our results imply that r(A) ∈ σ(A). It is classical [4], [8] that if K is a closed, total cone in a real Banach space X and A ∈ L(X) is a positive operator whose resolvent function R A (λ) has a pole at some λ 1 ∈ C with |λ 1 | = r(A), then r(A) ∈ σ(A); a transparent demonstration of this fact, differing radically from previous proofs, follows from our approximation-theoretic approach.…”

Section: Principal Question: Under What Further Conditions On a Is Itmentioning

confidence: 99%

See 2 more Smart Citations

Approximation by polynomials with nonnegative coefficients and the spectral theory of positive operators

Nussbaum

Walsh

1998

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. For Σ a compact subset of C symmetric with respect to conjugation and f : Σ → C a continuous function, we obtain sharp conditions on f and Σ that insure that f can be approximated uniformly on Σ by polynomials with nonnegative coefficients. For X a real Banach space, K ⊆ X a closed but not necessarily normal cone with K − K = X, and A : X → X a bounded linear operator with A[K] ⊆ K, we use these approximation theorems to investigate when the spectral radius r(A) of A belongs to its spectrum σ(A). A special case of our results is that if X is a Hilbert space, A is normal and the 1-dimensional Lebesgue measure of σ(i(A − A * )) is zero, then r(A) ∈ σ(A). However, we also give an example of a normal operator A = −U − αI (where U is unitary and α > 0) for which A[K] ⊆ K and r(A) / ∈ σ(A).

show abstract

A note on the Krein–Rutman theorem for sectorial operators

Wang

Zhou

2023

Mathematische Nachrichten

View full text Add to dashboard Cite

In this paper, we present some generalized versions of the Krein–Rutman theorem for sectorial operators. They are formulated in a fashion that can be easily applied to elliptic operators. Another feature of these generalized versions is that they contain some information on the generalized eigenspaces associated with nonprincipal eigenvalues, which are helpful in the study of the dynamics of evolution equations in ordered Banach spaces.

show abstract

Linear Operators in Complete Positive Cones

Abstract: THEOREM 1. If T is compact in X + and has non-zero partial spectral radius /A, then there exists a non-zero vector u in X+ such that Tu = pu.

Cited by 86 publications

References 2 publications

Positive operators and Hausdorff dimension of invariant sets

Positive operators and Hausdorff dimension of invariant sets

Approximation by polynomials with nonnegative coefficients and the spectral theory of positive operators

A note on the Krein–Rutman theorem for sectorial operators

Contact Info

Product

Resources

About