Invariant subspaces of weakly compact-friendly operators

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

We prove that if a non-zero weakly compact-friendly operator B on a Banach lattice with topologically full center is locally quasi-nilpotent, then the super right-commutant [B\rangle of B has a non-trivial closed invariant ideal. An example of a weakly compact-friendly operator which is not compact-friendly is also provided.

Anahtar Kelimeler:

Banach lattice, topologically full center, invariant subspace, weakly compact-friendly

Invariant subspaces of weakly compact-friendly operators

Keywords:

Banach lattice, topologically full center, invariant subspace, weakly compact-friendly,

PDF

___

Set S := RA2RA1R≥ 0. Since |π3π2π1x| ≤ S|x| for each x ∈ E , it follows that S = 0. Moreover, since each πi(i = 1, 2, 3) is dominated by a compact operator, we have by [2, Theorem 5.14] that π3π2π1is compact. Moreover, because R , A1, and A2belong to [B , so does S . Thus, [B contains a non-zero positive operator which dominates a compact operator. Now, invoke Theorem 2.2 to complete the proof. Abramovich, Y.A., Aliprantis, C.D.: An Invitation to Operator Theory. Graduate Studies in Mathematics. Vol. 50. Providence-RI. American Mathematical Society 2002.
Aliprantis, C.D., Burkinshaw, O.: Positive Operators. The Netherlands. Springer 2006.
C¸ a˘glar, M., Mısırlıo˘glu, T.: Weakly compact-friendly operators. Vladikavkaz Mat. Zh. 11(2), 27–30 (2009).
Flores, J., Tradacete, P., Troitsky, V.G.: Invariant subspaces of positive strictly singular operators on Banach lattices. J. Math. Anal. Appl. 343(2), 743–751 (2008).
Wickstead, A.W.: Extremal structure of cones of operators. Quart. J. Math. Oxford. 32(2), 239–253 (1981).
Wickstead, A.W.: Banach lattices with topologically full centre. Vladikavkaz Mat. Zh. 11(2), 50–60 (2009).
Mert C¸ A ˘GLAR, Tun¸c MISIRLIO ˘GLU Department of Mathematics and Computer Science, ˙Istanbul K¨ult¨ur University, Bakırk¨oy 34156, ˙Istanbul-TURKEY e-mail: t.misirlioglu@iku.edu.tr