@MISC{Djordjević_theproper, author = {Slaviša V. Djordjević}, title = {The Proper Elements and Simple Invariant Subspaces}, year = {} }

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Abstract

Abstract. A proper element of X is a triple (λ, L, A) composed by an eigenvalue λ, an invariant subspace of an operator A in B(X) generated by one eigenvector of λ and the operator A. For (λ0, L0,A0) ∈ Eig(X), where L0 = L ({x0}), the operator A0 induces an operator cA0 from the quotient X/L0 into itself, i.e.cA0(x + L0) = A0(x) + L0. In paper we show that λ0 is a simple pole of A0 if and only if λ0 / ∈ σ(cA0). Follow this concept we can define simple invariant subspaces of linear operator T like invariant subspace E such that σ(TE) ∩σ(cTE) =;, where TE: E → E is the restriction of T on E, cTE is the operator cTE (pi(y)) = pi(T(y)) on the quotient space X/E and pi is the natural homoeomorphism between X and X/E. Also, we give some properties of stability of simple invariant subspaces.