a-Weyl's theorem and the single valued extension property

Throughout this paper, ℒ(X) denotes the algebra of all linear bounded operators on an infinite-dimensional complex Banach space X and K(X) its ideal of compact operators. For an operator ℒ(X), write for its adjoint; () for its kernel; () for its range; () for its spectrum; ₐₚ() for its approximate point spectrum; ₛu() for its surjective spectrum and ₚ () for its point spectrum.peerReviewe

Linear mapping preserving the kernel or the range of operators

Let X and Y be two infinite dimensional real or complex Banach spaces. In this note we determine the forms of surjective additive maps : ℒ(X) ⟶ ℒ (Y) preserving the kernel's dimension or the range's codimension. As consequence, we establish that : ℒ (X) ⟶ℒ (X) preserves the kernel (respectively, the range) if and only if there exists an invertible operator A ∈ ℒ (X) such that (T) = AT (respectively, (T) = TA) for all T ∈ ℒ(X).peerReviewe

a-Weyl's theorem and the single valued extension property

In the present paper, we study a-Weyl's and a-Browder's theorem for an operator T such that T or T* satisfies the single valued extension property (SVEP). We establish that if T* has the SVEP, then T obeys a-Weyl's theorem if and only if it obeys Weyl's theorem. Further, if T or T* has the SVEP, we show that the spectral mapping theorem holds for the essential approximative point spectrum, and that a-Browder's theorem is satisfied by f(T) whenever f Î H(s(T)). We also provide several conditions that force an operator with the SVEP to obey a-Weyl's theorem. The author would like to precise that this paper constitute a part of his thesis [16]

Nonlinear maps preserving Drazin invertible operators of bounded index

Given an integer n ≥1, we provide a complete description of all bijective bicontinuous maps, on the algebra of all bounded linear operators acting on an innite-dimensional complex or real Banach space, that preserve the difference of Drazin invertible operators of index non-greater than n in both  directions. Key words: Nonlinear preserver problems, Drazin inverse, ascent,  descent

Nonlinear maps preserving Drazin invertible operators of bounded index

Given an integer n ≥1, we provide a complete description of all bijective bicontinuous maps, on the algebra of all bounded linear operators acting on an  innite-dimensional complex or real Banach space, that preserve the difference of Drazin invertible operators of index non-greater than n in both directions