Irreducible compact operators

Abstract

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