ON THE DECOMPOSITION OF OPERATORS WITH SEVERAL ALMOST-INVARIANT SUBSPACES

We seek a sufficient condition which preserves almost-invariant subspaces under the weak limit of bounded operators. We study the bounded linear operators which have a collection of almost-invariant subspaces and prove that a bounded linear operator on a Banach space, admitting each closed subspace as an almost-invariant subspace, can be decomposed into the sum of a multiple of the identity and a finite-rank operator.

Left-right fredholm and left-right browder linear relations

In this paper we introduce the notions of left (resp. right) Fredholm and left (resp. right) Browder linear relations. We construct a Kato-type decomposition of such linear relations. The results are then applied to give another decomposition of a left (resp. right) Browder linear relation T in a Banach space as an operator-like sum T = A + B, where A is an injective left (resp. a surjective right) Fredholm linear relation and B is a bounded finite rank operator with certain properties of commutativity. The converse results remain valid with certain conditions of commutativity. As a consequence, we infer the characterization of left (resp. right) Browder spectrum under finite rank operator.

On the stability of the spectral properties under commuting perturbations

In this paper, we examine the stability of several spectral properties under commuting perturbations. In particular, we show that if T ( L(X) is an isoloid operator satisfying generalized Weyl?s theorem and if F ( L(X) is a power finite rank operator that commutes with T, then generalized Weyl?s theorem holds for T + F. In addition, we consider the permanence of Bishop?s property (?), at a point, under commuting perturbation that is an algebraic operator.

FINITE RANK RIESZ OPERATORS

We provide conditions under which a Riesz operator defined on a Banach space is a finite rank operator.

Decomposability of finite rank operators in certain subspaces and algebras

Let  be either a reflexive subspace or a bimodule of a reflexive algebra in B (H), the set of bounded operators on a Hilbert space H. We find some conditions such that a finite rank T ∈  has a rank one summand in  and  has strong decomposability. Let (ℒ) be the set of all operators on H that annihilate all the operators of rank at most one in alg ℒ. We construct an atomic Boolean subspace lattice ℒ on H such that there is a finite rank operator T in (ℒ) such that T does not have a rank one summand in (ℒ). We obtain some lattice-theoretic conditions on a subspace lattice ℒ which imply alg ℒ is strongly decomposable.

Complexity of the Decidability of the Unquantified Set Theory with A Rank Operator

The unquantified set theory MLSR containing the symbols ∪, \, =, ∈, R (R(x) is interpreted as a rank of x) is considered. It is proved that there exists an algorithm which for any formula Q of the MLSR theory decides whether Q is true or not using the space c|Q|3 (|Q| is the length of Q).