Modulus Semigroups and Perturbation Classes for Linear Delay Equations in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper we study C₀-semigroups on X × L_p( − h, 0; X) associated with linear differential equations with delay, where X is a Banach space. In the case that X is a Banach lattice with order continuous norm, we describe the associated modulus semigroup, under minimal assumptions on the delay operator. Moreover, we present a new class of delay operators for which the delay equation is well-posed for p in a subinterval of [1,∞). Dedicated to the memory of H. H. Schaefer     

On the class of weak almost limited operators

Abstract

We introduce and study the class of weak almost limited operators. We establish a characterization of pairs of Banach lattices E, F for which every positive weak almost limited operator T : E→F is almost limited (resp. almost Dunford- Pettis). As consequences, we will give some interesting results.     

Compactness in Vector-valued Banach Function Spaces

We give a new proof of a recent characterization by Diaz and Mayoral of compactness in the Lebesgue-Bochner spaces L_X^p, where X is a Banach space and 1≤ p<∞, and extend the result to vector-valued Banach function spaces E_X, where E is a Banach function space with order continuous norm. The author is supported by the ‘VIDI subsidie’ 639.032.201 in the ‘Vernieuwingsimpuls’ programme of the Netherlands Organization for Scientific Research (NWO) and by the Research Training Network HPRN-CT-2002-00281.     

On principal T-bands in a Banach lattice

It is shown that for every positive order continuous Riesz operatorT, defined on an order complete complex Banach latticeE which is separated by its Köthe dual, there exists a Frobenius decomposition ofE into a countable number of disjoint principalT-bands and a band on whichT is quasi-nilpotent. A basis for the generalized eigenspace ofT pertaining to its maximal eigenvalue is constructed and the positivity properties of its elements are studied. The distinguished eigenvalues ofT are characterized and it is also shown that the theory ofT-bands is symmetric with respect to the duality which exists betweenE and its Köthe dual. This generalizes aspects of work done by H.D. Victory and R.-J. Jang-Lewis.     

A Dodds–Fremlin Property for Asplund and Radon–Nikodým Operators

Let E and F be Banach lattices and let S, T: E→ F be positive operators such that 0≤ S≤ T. It is shown that if T is a Radon–Nikodym operator, F has order continuous norm and E and F both have (Schaefer's) property (P), then S is a Radon–Nikodym operator; also, if T is an Asplund operator, E' has order continuous norm and E has property (P), then S is an Asplund operator.     

Isometric Equivalence of Certain Operators on Banach Spaces

A pair of operators on a Banach space X are isometrically equivalent if they are intertwined by a surjective isometry of X. We investigate the isometric equivalence problem for pairs of operators on specific types of Banach spaces. We study weighted shifts on symmetric sequence spaces, elementary operators acting on an ideal I of Hilbert space operators, and composition operators on the Bloch space. This last case requires an extension of known results about surjective isometries of the Bloch space.     

A lattice approach to narrow operators

It is known that if a rearrangement invariant function space E on [0,1] has an unconditional basis then each linear continuous operator on E is a sum of two narrow operators. On the other hand, the sum of two narrow operators in L₁ is narrow. To find a general approach to these results, we extend the notion of a narrow operator to the case when the domain space is a vector lattice. Our main result asserts that the set N_r(E, F) of all narrow regular operators is a band in the vector lattice L_r(E, F) of all regular operators from a non-atomic order continuous Banach lattice E to an order continuous Banach lattice F. The band generated by the disjointness preserving operators is the orthogonal complement to N_r(E, F) in L_r(E, F). As a consequence we obtain the following generalization of the Kalton-Rosenthal theorem: every regular operator T : E → F from a non-atomic Banach lattice E to an order continuous Banach lattice F has a unique representation as T = T_D + T_N where T_D is a sum of an order absolutely summable family of disjointness preserving operators and T_N is narrow. Supported by Ukr. Derzh. Tema N 0103Y001103.     

Geometric eigenspace of the generator ofC 0-semigroup of positive operators

LetE be a Banach lattice with order continuous norm and {T(t)} _t0 be an eventually compactc ₀-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.     

On the Property (gw)

In this note we introduce and study the property (gw), which extends property (w) introduced by Rakoc̆evic in [23]. We investigate the property (gw) in connection with Weyl type theorems. We show that if T is a bounded linear operator T acting on a Banach space X, then property (gw) holds for T if and only if property (w) holds for T and Π _a (T) = E(T), where Π _a (T) is the set of left poles of T and E(T) is the set of isolated eigenvalues of T. We also study the property (gw) for operators satisfying the single valued extension property (SVEP). Classes of operators are considered as illustrating examples. The second author was supported by Protars D11/16 and PGR- UMP.     

Some remarks on disjointness preserving operators

In this note we present a simple proof of the following results: if T: E E is a lattice homomorphism on a Banach lattice E, then: i) (T)={1} implies T=I; and ii) r(T–I)<1 implies TZ(E), the center of E.     

Algebraic reflexivity of some subsets of the isometry group

Let X be a compact first countable space. In this paper we show that the set of isometries of C(X) that are involutions is algebraically reflexive. As a consequence of a recent work of Botelho and Jamison this leads to the conclusion that the set of generalized bi-circular projections on C(X) is also algebraically reflexive. We also consider these questions for the space C(X,E) where E is a uniformly convex Banach space.     

Weakly equicompact sets of operators defined on Banach spaces

Let X and Y be Banach spaces. We say that a set (the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (x_n) in X, there exists a subsequence (x_k(n)) so that (Tx_k(n)) is weakly uniformly convergent for T ∈ M. We study some properties of weakly equicompact sets and, among other results, we prove: 1) if is collectively weakly compact, then M^* is weakly equicompact iff M^** x^**={T^** x^** : T ∈ M} is relatively compact in Y for every x^** ∈X^**; 2) weakly equicompact sets are precompact in for the topology of uniform convergence on the weakly null sequences in X. Received: 14 February 2005; revised: 1 June 2005     

Invariant subspaces and localizable spectrum

LetT L(X) be a continuous linear operator on a complex Banach spaceX. We show thatT possesses non-trivial closed invariant subspaces if its localizable spectrum _loc(T) is thick in the sense of the Scott Brown theory. Since for quotients of decomposable operators the spectrum and the localizable spectrum coincide, it follows that each quasiaffine transformation of a Banach-space operator with Bishop's property () and thick spectrum has a non-trivial invariant subspace. In particular it follows that invariant-subspace results previously known for restrictions and quotients of decomposable operators are preserved under quasisimilarity.     

SOME REMARKS CONCERNING THE DAUGAVET EQUATION

Abstract

We say that a normed space X has the Daugavet property (DP) if for every finite rank operator K in X the equality ∥I + T∥ = 1 + ∥T∥ holds. It is known that C[0,1] and L ₁[0,1] have DP. We prove that if X has DP then X has no unconditional basis. We also discuss anti-Daugavet property, hereditary DP-spaces and construct a strictly convex normed space having DP.     

On the approximation of positive operators and the behaviour of the spectra of the approximants

LetT be a positive linear operator on the Banach latticeE and let (S _n ) be a sequence of bounded linear operators onE which converge strongly toT. Our main results are concerned with the question under which additional assumptions onS _n andT the peripheral spectra (S _n ) ofS _n converge to the peripheral spectrum (T) ofT. We are able to treat even the more general case of discretely convergent sequences of operators.     

On tree characterizations of <Emphasis Type="Italic">G</Emphasis><Subscript>δ</Subscript>-embeddings and some Banach spaces

We show that a one-to-one bounded linear operator T from a separable Banach space E to a Banach space X is a G _δ-embedding if and only if every T-null tree in S _E has a branch which is a boundedly complete basic sequence. We then consider the notions of regulators and skipped blocking decompositions of Banach spaces and show, in a fairly general set up, that the existence of a regulator is equivalent to that of special skipped blocking decomposition. As applications, the following results are obtained. (a) A separable Banach space E has separable dual if and only if every w*-null tree in S _E * has a branch which is a boundedly complete basic sequence. (b) A Banach space E with separable dual has the point of continuity property if and only if every w-null tree in S _E has a branch which is a boundedly complete basic sequence. We also give examples to show that the tree hypothesis in both the cases above cannot be replaced in general with the assumption that every normalized w*-null (w-null in (b)) sequence has a subsequence which is a boundedly complete basic sequence. The research of S. Dutta was supported in part by the Institute for Advanced Studies in Mathematics at Ben-Gurion University of the Negev. The research of V. P. Fonf was supported in part by Israel Science Foundation, Grant No. 139/03.     

On nonnegative solvability of linear operator equations

LetE be a Banach lattice having order continuous norm. Suppose, moreover,T is a nonnegative reducible operator having a compact iterate and which mapsE into itself. The purpose of this work is to extend the previous results of the authors, concerning nonnegative solvability of (kernel) operator equations on generalL ^p-spaces. In particular, we provide necessary and sufficient conditions for the operator equation x=T x+y to possess a nonnegative solutionxE wherey is a given nonnegative and nontrivial element ofE and is any given positive parameter.     

On the Decomposition of the Riesz Operator and the Expansion of the Riesz Semigroup

For a Riesz operator T on a reflexive Banach space X with nonzero eigenvalues denote by E(λ_i; T) the eigen-projection corresponding to an eigenvalue λ_i. In this paper we will show that if the operator sequence is uniformly bounded, then the Riesz operator T can be decomposed into the sum of two operators T_p and T_r: T = T_p + T_r, where T_p is the weak limit of T_n and T_r is quasi-nilpotent. The result is used to obtain an expansion of a Riesz semigroup T(t) for t ≥ τ. As an application, we consider the solution of transport equation on a bounded convex body.     

Some results on b-AM-compact operators

We show that for positive operator B : E → E on Banach lattices, if there exists a positive operator S : E → E such that:1.SB ≤ BS;2.S is quasinilpotent at some x₀ > 0; 3.S dominates a non-zero b-AM-compact operator, then B has a non-trivial closed invariant subspace. Also, we prove that for two commuting non-zero positive operators on Banach lattices, if one of them is quasinilpotent at a non-zero positive vector and the other dominates a non-zero b-AM-compact operator, then both of them have a common non-trivial closed invariant ideal. Then we introduce the class of b-AM-compact-friendly operators and show that a non-zero positive b-AM- compact-friendly operator which is quasinilpotent at some x₀ > 0 has a non-trivial closed invariant ideal.