Finite Rank Operators Leaving Double Triangles Invariant (original) (raw)

Invariant subspaces of positive strictly singular operators on Banach lattices

Journal of Mathematical Analysis and …, 2008

It is shown that every positive strictly singular operator T on a Banach lattice satisfying certain conditions is AM-compact and has invariant subspaces. Moreover, every positive operator commuting with T has an invariant subspace. It is also proved that on such spaces the product of a disjointly strictly singular and a regular AM-compact operator is strictly singular. Finally, we prove that on these spaces the known invariant subspace results for compact-friendly operators can be extended to strictly singular-friendly operators.

SOME PROPERTIES OF STRICTLY SINGULAR OPERATORS ON BANACH LATTICES

maia.ub.es

Several results obtained during the author's Ph.D. Thesis are presented. In particular, domination results (in Dodds-Fremlin sense) for the ideal of strictly singular operators will be given. Moreover, the connections between strictly singular and the classes of AM-compact, l2-singular and disjointly strictly singular are studied. As an application we obtain existence of invariant subspaces for positive strictly singular operators. On a di erent direction, results on compact powers of strictly singular operators are also presented extending a theorem of V. Milman. Finally, we study when a c0-singular or l1-singular operator can be extended to an operator between vector valued lattices preserving its singularity properties.

Disjointly non-singular operators on Banach lattices

Journal of Functional Analysis, 2021

An operator T from a Banach lattice E into a Banach space is disjointly non-singular (DN-S, for short) if no restriction of T to a subspace generated by a disjoint sequence is strictly singular. We obtain several results for DN-S operators, including a perturbative characterization. For E = L p (1 < p < ∞) we improve the results, and we show that the DN-S operators have a different behavior in the cases p = 2 and p = 2. As an application we prove that the strongly embedded subspaces of L p form an open subset in the set of all closed subspaces.

Triangles Which are Bounded Operators on Ak

2000

A lower triangular infinite matrix is called a triangle if there are no zeros on the principal diagonal. The main result of this paper gives a minimal set of sucient conditions for a triangle T : Ak ! Ak for the sequence space

Invariant subspace lattices for a class of operators

Pacific Journal of Mathematics, 1981

We study the invariant subspace lattices for a one parameter family of operators {T a } a on L p (0,1), a a complex number, where Taf(x) = xf(x) + a [ X f(t)dt , Jo and their adjoints TJ, T*f(x)xf(x) + a [f(t)dt. The closed invariant subspaces for T a are in one-to-one correspondence with certain closed ideals of & a9 where & a is a Silov algebra with unit and in which the range & a of the Riemann Liouville operator J a Γ(a) is embedded as a closed ideal. When n is a positive integer, there is a complete lattice isomorphism between the closed ideals of & n and the ^-tuples (E Of E lf , E nλ) of closed subsets of [0, 1] where E o Ώ E± Ώ Ώ. E n-X Ώ. derived set of E o. Every closed ideal of & n is the intersection of closed primary ideals. Similar results carry over to a where the real part of a is an integer and also to the adjoint operators.

Invariant Subspaces for Semigroups of Algebraic Operators

Journal of Functional Analysis, 1998

, 269 305) that a semigroup of matrices is triangularizable if the ranks of all the commutators of elements of the semigroup are at most 1. Our main theorem is an extension of this result to semigroups of algebraic operators on a Banach space. We also obtain a related theorem for a pair [A, B] of arbitrary bounded operators satisfying rank (AB&BA)=1 and several related conditions. In addition, it is shown that a semigroup of algebraically unipotent operators of bounded degree is triangularizable. 1998 Academic Press Definition. A collection of bounded linear operators on a complex Banach space is triangularizable if there is a chain of subspaces which is article no. FU983293 452 0022-1236Â98 25.00

c 0-Singular and ℓ 1-singular operators between vector-valued Banach lattices

Positivity

Given an operator T : X --> Y between Banach spaces, and a Banach lattice E consisting of measurable functions, we consider the point-wise extension of the operator to the vector-valued Banach lattices TE : E(X) --> E(Y ) given by TE(f)(x) = T(f(x)). It is proved that for any Banach lattice E which does not contain c0, the operator T is an isomorphism on a subspace isomorphic to c0 if and only if so is TE. An analogous result for invertible operators on subspaces isomorphic to l1 is also given.