On the ring ideal generated by a positive operator

Let Z be a Banach lattice endowed with positive cone C and an order-continuous norm j.j .Let G be a left-amenable semigroup of positive linear endomorphisms of Z .Then the positive fixed points Co of Z under G form a lattice cone, and their linear span Z0 is a Banach lattice under an order-continuous norm ||.||0 which agrees with |.| on Co.A counterexample shows that under the given conditions Z0 need not contain all the fixed points of Z under G , and need not be a sublattice of (Z, C).The paper concludes with a discussion of some related results.Let G be a semigroup.We denote by m(C7) the Banach space of all bounded linear functions from G into the real numbers R, under the supremum norm.We denote by m'iG) the Banach dual of w(C7).With each T g G we associate an endomorphism Tm of /n(C7) defined by iTmb)iU) = biTU) for U G G and b G m(G) where TU denotes the composite of T and U under the semigroup operation.An element p G m*iG) is called a mean for G iff inf biT) < pib) < sup biT) for all b G G TG TG and left-invariant for G iff T'mp=p forallTGG, where T'm denotes the adjoint of Tm .Following M. Day [1, p. 108] we call the semigroup G left-amenable iff there exists a left-invariant mean for G.In particular, any Abelian semigroup is left-amenable [1, Theorem 4, p. 108].A Banach lattice is said to have order-continuous norm iff every decreasing sequence of positive elements is norm convergent [3, 5.12, p. 92; 5.10(d), p. 89].Theorem.Let (Z,C, ||.||) be a Banach lattice with order-continuous norm.Let G be a left-amenable semigroup of positive linear operators from Z into Z.Define C0 = {x G C : Tx = x for all T gG} , Z0 = C0 -CQ.

Über die konvergenz positiver operatoren mit werten in atomaren Banachverbänden mit ordnungsstetiger norm

In terms of quasi-interior points, criteria that a Banach lattice has order continuous norm or is an -space with a unit are given. For example, if is Dedekind complete and has a weak order unit, then has order continuous norm if and only if the set of quasi-interior points of coincides with the set of weak order units of ; a Banach lattice is an -space with a unit if and only if the set of all quasi-interior points of coincides with the set { : for some > 0}. Analogous questions are considered for the case of an ordered Banach space with a cone . Moreover, it is shown that every nonzero point of a cone = {0} is quasi-interior if and only if dim = 1. We also study various D-properties of a cone ; in particular, the conditions for which the relation = 0 with > 0 implies that is not a quasi-interior point are considered.

We give a characterization of those cr-Dedekind complete Banach lattices for which every continuous linear operator T: E -> co is a difference of two positive linear operators from E into cq .

Throughout this paper we denote by L p the Banach lattice of p-integrable functions on a σ-finite measure space (X, B, μ), where 1 ≤ p ≤ ∞. We will consider those aspects of the theory of positive linear operators, which are in some way special due to the fact the operators are acting on L p-spaces. For general information about positive operators on Banach lattices we refer to the texts [1]. [20], and [36]. Our focus on L p-spaces does not mean that in special cases some of the results can not be extended to a larger class of Banach lattices of measurable function such as Orlicz spaces or re-arrangement invariant Banach function spaces. However in many cases the results in these extensions are not as precise or as complete as in the case of L p-spaces. We will discuss results related to the boundedness of positive linear operators on L p-spaces. The most important result is the so-called Schur criterion for boundedness. This criterion is the most frequently used tool to show that a concrete positive linear operator is bounded from L p to L q. Then we will show how this result relates to the change of density result of Weis [33]. Next the equality case of Schur’s criterion is shown to be closely related to the question whether a given positive linear operator attains its norm. We discuss in detail the properties of norm attaining operators on L p-spaces and discuss as an example the weighted composition operators on L p-spaces. Then we return to the Schur criterion and show how it can be applied to the factorization theorems of Maurey and Nikisin. Most results mentioned in this paper have appeared before in print, but sometimes only implicitly and scattered over several papers. Also a number of the proofs presented here are new.

R. Drnovšek, D. Kokol-Bukovšek, L. Livshits, G. MacDonald, M. Omladič, and H. Radjavi constructed an irreducible set of positive nilpotent operators on $L^p[0,1)$ which is closed under multiplication, addition and multiplication by positive real scalars with the property that any finite subset is ideal-triangularizable. In this paper we prove the following: every algebra of nilpotent operators which is generated by a set of positive operators on a Banach lattice is ideal-triangularizable whenever the nilpotency index of its operators is bounded; every finite subset of an algebra of nilpotent operators which is generated by a set of positive operators on a Banach lattice is ideal-triangularizable.

Chapter 2 Positive operators

We show that even in the case of a Banach lattice E with an order continuous norm (or whose dual has an order continuous norm) the linear span of the positive compact operators on E need not be complete under the regular norm.

For Banach lattices E and F, let P, Q : E → F be n-homogeneous polynomials such that 0 ≤ Q ≤ P. We prove that if F has an order continuous norm and P is weakly compact then Q is also weakly compact. We also prove that if F is an atomic Banach lattice with an order continuous norm and P is compact then Q is also compact.

Unbounded order convergence in dual spaces

Let E be a Banach lattice, T be a bounded operator on E. The Weyl essential spectrum σew(T) of the operator T is a set $$\sigma_{\rm ew}(T) = {\mathop{\cap}\limits_{K \in {\mathcal {K}}(E)}} \sigma (T + K)$$ , where $${\mathcal K}(E)$$ is a set of all compact operators on E. In particular for a positive operator T next subsets of the spectrum $$\sigma_{\rm ew}^+(T) = \bigcap\limits_{0 \le K \in {\mathcal K}(E)} \sigma (T + K), \ \ \ \ \ \sigma_{\rm ew}^-(T) = \bigcap\limits_{0 \le K \in {\mathcal K}(E) \le T} \sigma (T - K)$$ are introduced in the article. The conditions by which $$r(T) \not\in \sigma_{\rm ef}(T)$$ implies either $$r(T) \not\in \sigma_{\rm ew}^+(T)$$ or $$r(T) \not\in \sigma_{\rm ew}^-(T)$$ are investigated, where σef(T) is the Fredholm essential spectrum. By this reason, the relations between coefficients of the main part of the Laurent series of the resolvent R(., T) of a positive operator T around of the point λ = r(T) are studied. The example of a positive integral operator T : L1→ L∞ which doesn’t dominate a non-zero compact operator, is adduced. Applications of results which are obtained, to the spectral theory of band irreducible operators, are given. Namely, the criteria when the operator inequalities 0 ≤ S < T imply the spectral radius inequality r(S) < r(T), are established, where T is a band irreducible abstract integral operator.

In this paper we study the structure of the set {\rm Hom}(X,\mathbb{R}) of all lattice homomorphisms from a Banach lattice X into \mathbb{R} . Using the relation among lattice homomorphisms and disjoint families, we prove that the topological dual of the free Banach lattice {\rm FBL}(A) generated by a set A contains a disjoint family of cardinality 2^{|A|} , answering a question of B. de Pagter and A. W. Wickstead. We also deal with norm-attaining lattice homomorphisms. For classical Banach lattices, as c_0 , L_p - and C(K) -spaces, every lattice homomorphism on it attains its norm, which shows, in particular, that there is no James theorem for this class of functions. We prove that, indeed, every lattice homomorphism on X and C(K,X) attains its norm whenever X has order continuous norm. On the other hand, we provide what seems to be the first example in the literature of a lattice homomorphism which does not attain its norm. In general, we study the existence and characterization of lattice homomorphisms not attaining their norm in free Banach lattices. As a consequence, it is shown that no Bishop–Phelps type theorem holds true in the Banach lattice setting, i.e., not every lattice homomorphism can be approximated by norm-attaining lattice homomorphisms.

An operator T from a Banach space E into a Banach lattice F is said to be semi-compact if for each e > 0, there exists some u ∈ F+ such that T (BE) ⊂ [−u, u] + eBF where BH is the closed unit ball of H = E or F and F+ = {x ∈ F : 0 ≤ x}. Contrarily to compact operators, the class of semi-compact operators satisfies the domination problem. Indeed, if S and T are operators from a Banach lattice E into F such that 0 ≤ S ≤ T and T is semi-compact, then S is semi-compact (Theorem 18.20 of [3]). It is clear that each compact operator is semi-compact but a semicompact operator is not necessary compact. In fact, the operator identity Idl∞ : l∞ −→ l∞ is semi-compact but it is not compact. Let us recall that Dodds and Fremlin ([11], Theorem 125.5) proved that if E and F are two Banach lattices such that the norms of F and of the topological dual E ′ are order continuous, then each semi-compact operator T from E into F which is AM-compact (i.e. a regular operator such that the image of each order bounded subset of E is a relatively compact subset of F ) is compact. In [5], we studied the compactness of positive Dunford-Pettis operators, and we gave several characterizations of Banach lattices on which each

We give conditions for the linear span of the positive L-weakly compact (resp. M-weakly compact) operators to be a Banach lattice under the regular norm, for that Banach lattice to have an order continuous norm, to be an AL-space or an AM-space.

LetE be a Banach lattice with order continuous norm and {T(t)}t≥0 be an eventually compactc0-semigroup of positive operators onE with generator A. We investigate the structure of the geometric eigenspace of the generator belonging to the spectral bound when the semigroup is ideal reducible. It is shown that a basis of the eigenspace can be chosen to consist of element ofE with certain positivity structure. This is achieved by a decomposition of the underlying Banach latticeE into a direct sum of closed ideals which can be viewed as a generalization of the Frobenius normal form for nonnegative reducible matrices.