Related papers: On Almost-Invariant Subspaces and Approximate Comm…

Given a Banach space X and a bounded linear operator T on X, a subspace Y of X is almost invariant under T if TY is a subspace of Y+F for some finite-dimensional ``error'' F. In this paper, we study subspaces that are almost invariant under…

We introduce and study the following modified version of the Invariant Subspace Problem: whether every operator T on a Banach space has an almost invariant half-space, that is, a subspace Y of infinite dimension and infinite codimension…

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…

If $T$ is a bounded linear operator acting on an infinite-dimensional Banach space $X$, we say that a closed subspace $Y$ of $X$ of both infinite dimension and codimension is an almost-invariant halfspace (AIHS) under $T$ whenever…

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…

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 }…

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.…

For a shift operator $T$ with finite multiplicity acting on a separable infinite dimensional Hilbert space we represent its nearly $T^{-1}$ invariant subspaces in terms of invariant subspaces under the backward shift. Going further, given…

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…

In this paper, we study {\it operator spaces\/} in the sense of the theory developed recently by Blecher-Paulsen [BP] and Effros-Ruan [ER1]. By an operator space, we mean a closed subspace $E\subset B(H)$, with $H$ Hilbert. We will be…

For an operator $A$ on a complex Banach space $X$ and a closed subspace $M\subseteq X$, the local commutant of $A$ at $M$ is the set $C(A;M)$ of all operators $T$ on $X$ such that $TAx=ATx$ for every $x\in M$. It is clear that $ C(A;M)$ is…

It is proved that a commutative algebra $A$ of operators on 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 }…

In this paper we formulate the almost invariant subspaces theorems of backward shift operators in terms of the ranges or kernels of product of Toeplitz and Hankel operators. This approach simplifies and gives more explicit forms of these…

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…

An algebra A of operators on a Banach space X is called strictly semi-transitive if for all non-zero x,y in X there exists an operator S in A such that Sx=y or Sy=x. We show that if A is norm-closed and strictly semi-transitive, then every…

Let X be a complex Banach space of dimension at least 2, and let S be a multiplicative semigroup of operators on X such that the rank of AB - BA is at most 1 for all pairs {A,B} in S. We prove that S has a non-trivial invariant subspace…

It is shown that if the Deddens algebra ${\mathcal D}_T$ associated with a quasinilpotent operator $T$ on a complex Banach space is closed and localizing then $T$ has a nontrivial closed hyperinvariant subspace.

We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e. spaces $X$ such that the norm equality $\|Id + T^2\|=1 + \|T^2\|$ holds for every bounded linear operator $T:X\longrightarrow X$. This answers in the…

In this paper we completely characterize the norm attainment set of a bounded linear operator on a Hilbert space. This partially answers a question raised recently in [\textit{D. Sain, On the norm attainment set of a bounded linear…