Real Banach Lattices Research Articles

Invariant subspace problem for positive L-weakly and M-weakly compact operators

In this paper, we show that positive L-weakly and M-weakly compact operators on some real Banach lattices have a non-trivial closed invariant subspace. Also, we prove that any positive L-weakly (or M-weakly) compact operator T:E \rightarrow E\ has a non-trivial closed invariant subspace if there exists a Dunford-Pettis operator S:E \rightarrow E satisfying 0 \leq T \leq S, where E is Banach lattice.

An elementary approach to a lattice-valued Banach-Stone Theorem

Let $ X $ and $ Y $ be compact Hausdorff spaces, and $ E $ be a nonzero real Banach lattice. In this note, we give an elementary proof of a lattice-valued Banach-Stone theorem by Cao, Reilly and Xiong [3] which asserts that if there exists a Riesz isomorphism $ \Phi: C(X,E)\rightarrow C(Y,\mathbb{R}) $ such that $ \Phi(f) $ has no zeros if $ f $ has none, then $ X $ is homeomorphic to $ Y $ and $ E $ is Riesz isomorphic to $ \mathbb{R} $.

Order and algebra isomorphisms of spaces of regular operators

If E and F are real Banach lattices and there is an algebra and order isomorphism Φ:Open image in new window(E) →Open image in new window(F) between their respective ordered Banach algebras of regular operators then there is a linear order isomorphism U:E→F such that Φ(T) =UTU−1 for all T ∈ Open image in new window(E).

Monotonicity and complex convexity in Banach lattices

The goal of this article is to study the relations among monotonicity properties of real Banach lattices and the corresponding convexity properties in the complex Banach lattices. We introduce the moduli of monotonicity of Banach lattices. We show that a Banach lattice E is uniformly monotone if and only if its complexification E C is uniformly complex convex. We also prove that a uniformly monotone Banach lattice has finite cotype. In particular, we show that a Banach lattice is of cotype q for some 2 ⩽ q < ∞ if and only if there is an equivalent lattice norm under which it is uniformly monotone and its complexification is q-uniformly PL-convex. We also show that a real Köthe function space E is strictly (respectively uniformly) monotone and a complex Banach space X is strictly (respectively uniformly) complex convex if and only if Köthe–Bochner function space E ( X ) is strictly (respectively uniformly) complex convex.

The Norm of a Complex Banach Lattice

In this note we study the problem how the complexification of a real Banach space can be normed in such a way that it becomes a complex Banach space from the point of view of the theory of cross-norms on tensor products of Banach spaces. In particular we show that the norm of a complex Banach lattice can be interpretated in terms of the l-tensor product of real Banach lattices.

Canonical embedding of function spaces into the topological bidual of {$C(K;E)$}

Let K be a Hausdor compact space and E be a real Banach lattice with order continuous norm. In this paper, we essentially prove the existence of canonical embeddings of (vector) sublattices of FB(K;E), the Banach lattice of E valued bounded functions on K ,i nto the topological bidual of C(K;E), the usual Banach lattice of E valued continuous functions on K. This is related and extends some results in the real case of H. H. Schaefer ( [14], [15]).

A Characterization of Positive Constrictive Operators

In this note we prove that if T is a positive operator on a real Banach lattice, then T is constrictive if and only if that T has the operator matrix decomposition T(T1 0) -0 T2J where T1 is a power-bounded generalized permutation matrix on a finitedimensional space and T2n -* 0 strongly.

A characterization of positive constrictive operators

In this note we prove that if T is a positive operator on a real Banach lattice, then T is constrictive if and only if that T has the operator matrix decomposition \[ T = ( T 1 a m p ; 0 0 a m p ; T 2 ) , T = \left ( {\begin {array}{*{20}{c}} {{T_1}} &amp; 0 \\ 0 &amp; {{T_2}} \\ \end {array} } \right ), \] where T 1 {T_1} is a power-bounded generalized permutation matrix on a finite-dimensional space and T 2 n → 0 T_2^n \to 0 strongly.

On the “zero-two” law for positive contractions

Let (X, Σ,μ) be a measure space (where μ is a positive σ-additive measure) and let Lp(X,Σ,μ), 1≦p≦ + ∞ be the usual real Banach lattices.

Irreducible compact operators

The purpose of the present paper is to show that a positive compact (ideal-) irreducible operator T in a Banach lattice (of dimension greater than one) has spectral radius r (T)>0 , i.e., T is not quasi-nilpotent. It was shown in [6] that there exist positive irreducible operators which are quasi-nilpotent and the question under what conditions a positive irreducible operator T satisfies r ( T ) > 0 has been studied extensively (see e.g. [6] and [7], Sect. V.6). In particular we mention the Ando-Krieger theorem (see e.g. [9], Theorem 136.9), which states that a positive irreducible kernel operator T in a Banach function space has r (T)>0. Recently in [8], some special situations are discussed in which a compact positive irreducible operator has a strictly positive spectral radius. Let L be a (real or complex) Banach lattice. By 2~~ we denote the Banach space of all bounded linear operators in L. As usual, we write S__< T in ~ ( L ) whenever Su<Tu for all O<uEL. Given 0 < T ~ ( L ) , the closed ideal J in L is called T-invariant if T(J)~J, and T is called (ideal-) irreducible if the only closed T-invariant ideals are {0} and L (see [7], III.8). For the general theory of Banach lattices and positive operators we refer to the books [7] and [9]. In the proof of the main result of the paper we will use some properties of the center of a Banach lattice. We recall some elementary facts. Let L be a real Banach lattice. The center Z(L) of L is the subspace of 5~(L) consisting of all operators ~z for which there exists 0__<2~IR such that ] ~ f l < 2 ] f ] for all f~L. The center has the structure of a vector lattice with (nl v n2)u=(n l u)v (n2u) and OzlA~zz)u=(~lU)A(nzU ) for all O<u~L and all ~l, n2eZ(L). Moreover, Z(L) is a commutative algebra (with respect to composition as multiplication) and the identity operator I in L is a strong unit, i.e., for any rc~Z(L) there exists 0__<2EIR such that [~zI__<2I (see [9], Chap. 20). Note that if L = C(K) for some compact Hausdorff space K, then Z(L) can be identified with C(K) acting on itself by multiplication. If L is a Dedekind complete Banach lattice and 0_<u_< v in L, then it is an easy consequence of the Freudenthal spectral theorem ([4], Theorem 40.2) that there exists n~Z(L) such that zcv=u and O<_n<I. However, in an arbitrary Banach lattice such a 7z need not exist, as is shown by the Banach lattice L = C([0, 1]). The next lemma shows that, under rather mild conditions, we can

Fixräume regulärer operatoren und ein Korovkin-Satz

S. J. Bernau has introduced the notion of an exchange subspace of an L p -space and has shown that the range of a contractive linear projection on an L p -space (1 ⩽ p < ∞, p ≠ 2) is an exchange subspace. In the present paper we define this notion for real Banach lattices with order continuous norm and prove among other things that fixed spaces of special regular operators on these spaces are exchange subspaces. As application we give a Korovkin theorem for sequences of contractions on real Banach lattices with an uniformly monotone norm.