相关论文: Essential Spectra of Induced Operators on Subspace…

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 $\alpha\in\mathbb{C}$, $\alpha\neq 0$, $\alpha\neq…

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…

Let $T$ be an absolutely continuous polynomially bounded operator, and let $\theta$ be a singular inner function. It is shown that if $\theta(T)$ is invertible and some additional conditions are fulfilled, then $T$ has nontrivial…

We study linear operators $T$ on Banach spaces for which there exists a $C_0$-semigroup $(T(t))_{t\geq 0}$ such that $T=T(1)$. We present a necessary condition in terms of the spectral value 0 and give classes of examples where this can or…

For a power bounded or polynomially bounded operator $T$ sufficient conditions for the existence of a nontrivial hyperinvariant subspace are given. The obtained hyperinvariant subspaces of $T$ have the form of the closure of the range of…

In this paper we study sufficient conditions for an operator to have an almost-invariant half-space. As a consequence, we show that if $X$ is an infinite-dimensional complex Banach space then every operator $T\in\mathcal{L}(X)$ admits an…

We show that any bounded operator $T$ on a separable, reflexive, infinite-dimensional Banach space $X$ admits a rank one perturbation which has an invariant subspace of infinite dimension and codimension. In the non-reflexive spaces, we…

In contrast to the classical complex spectral theory, where the spectrum is related to the invertibility of $\lambda-A:D(A)\subseteq X_\mathbb{C}\rightarrow X_\mathbb{C}$, in the noncommutative quaternionic $S$-spectral theory one uses the…

A bounded operator $T$ in a Banach space $X$ is said to satisfy the essential descent spectrum equality, if the descent spectrum of $T$ as an operator on $X$ coincides with the essential descent spectrum of $T$. In this note, we give some…

If T is a bounded linear operator acting on an infinite-dimensional Banach space, then there exists and operator F of rank at most one and arbitrarily small norm such that T-F has an invariant subspace of infinite dimension and codimension.…

Let $E$ be a complex Banach lattice and $T$ is an operator in the centrum $Z(E)=\{T: |T|\le \lambda I \mbox{ for some } \lambda\}$ of $E$. Then the essential norm $\|T\|_{e}$ of $T$ equals the essential spectral radius $r_{e}(T)$ of $T$. We…

In this note we deal with the problem of determining the essential spectrum of a Toeplitz operator $T_{f}:A^{2}(\mathbb{D}^{2})\rightarrow A^{2}(\mathbb{D}^{2})$ acting on the Bergman space $A^{2}(\mathbb{D}^{2})$ of the bi-disc whose…

We construct an indecomposable reflexive Banach space $X_{ius}$ such that every infinite dimensional closed subspace contains an unconditional basic sequence. We also show that every operator $T\in \mathcal{B}(X_{ius})$ is of the form…

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…

It is proved that a commutative algebra $A$ of operators in a reflexive real Banach space has an invariant subspace if each operator $T\in A$ satisfies the condition $$\|1- \varepsilon T^2\|_e \le 1 + o(\varepsilon) \text{ when }…

We describe essential (in particular Fredholm and semi-Fredholm) spectra of operators on Banach lattices of the form $T=wU$, where $w$ is a central operator and $U$ is a disjointness preserving operator such that its spectrum $\sigma(U)$ is…

We investigate properties of essential spectra of disjointness preserving operators acting on Banach $C(K)$-modules. In particular, we prove that under some very mild conditions the upper semi-Fredholm spectrum of such an operator is…

A bounded linear operator $T$ on a Banach space $X$ is called subspace-hypercyclic if there is a subspace $M \subsetneq X$ and a vector $x \in X$ such that $orb{(x,T)} \cap M$ is dense in $M$. We show that every Banach space supports…

A bounded linear operator $T$ on a Banach space $X$ is called frequently hypercyclic if there exists $x\in X$ such that the lower density of the set $\{n\in\N:T^nx\in U\}$ is positive for any non-empty open subset $U$ of $X$. Bayart and…

Let ${\bf R}$ denote any of the following classes: upper (lower) semi-Fredholm operators, Fredholm operators, upper (lower) semi-Weyl operators, Weyl operators, upper (lower) semi-Browder operators, Browder operators. For a bounded linear…