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تسجيل مستخدم جديدOn quasinilpotent operators and the invariant subspace problem
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تأليف
Adi Tcaciuc
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We show that a bounded quasinilpotent operator $T$ acting on an infinite dimensional Banach space has an invariant subspace if and only if there exists a rank one operator $F$ and a scalar $alphainmathbb{C}$, $alpha eq 0$, $alpha eq 1$, such that $T+F$ and $T+alpha F$ are also quasinilpotent. We also prove that for any fixed rank-one operator $F$, almost all perturbations $T+alpha F$ have invariant subspaces of infinite dimension and codimension.
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We show that for any bounded operator $T$ acting on an infinite dimensional Banach space there exists an operator $F$ of rank at most one such that $T+F$ has an invariant subspace of infinite dimension and codimension. We also show that whenever the
boundary of the spectrum of $T$ or $T^*$ does not consist entirely of eigenvalues, we can find such rank one perturbations that have arbitrarily small norm. When this spectral condition is not satisfied, we can still find suitable finite rank perturbations of arbitrarily small norm, but not necessarily of rank one.
We show that for any bounded operator $T$ acting on infinite dimensional, complex Banach space, and for any $varepsilon>0$, there exists an operator $F$ of rank at most one and norm smaller than $varepsilon$ such that $T+F$ has an invariant subspace
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