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We show that finitely subgraded Lie algebras of compact operators have invariant subspaces when conditions of quasinilpotence are imposed on certain components of the subgrading. This allows us to obtain some useful information about the…

We extend the method of minimal vectors to arbitrary Banach spaces. It is proved, by a variant of the method, that certain quasinilpotent operators on arbitrary Banach spaces have hyperinvariant subspaces.

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…

A Banach space contains either a minimal subspace or a continuum of incomparable subspaces. General structure results for analytic equivalence relations are applied in the context of Banach spaces to show that if $E_0$ does not reduce to…

Let $T$ be a quasinilpotent operator on a Banach space. Under assumptions of a certain nonsymmetry in the growth of the resolvent of $T$, it is proved that every operator in the commutant of $T$ is not unicellular. In particular, $T$ has…

The linear isometries between weighted Banach spaces of continuous functions are considered. Some of well known theorems on isometries between spaces of continuous functions are proved and stated, but all they are in an appropriate form. In…

We present partial results to the following question: Does every infinite dimensional Banach space have an infinite dimensional subspace on which one can define an operator which is not a compact perturbation of a scalar multiplication?

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…

In the setting of operators on Hilbert spaces, we prove that every quasinilpotent operator has a non-trivial closed invariant subspace if and only if every pair of idempotents with a quasinilpotent commutator has a non-trivial common closed…

In the paper we extend the spectral invariance of pseudodifferential operators acting on (non-weighted) classical modulation spaces to allow the Lebesgue exponents to be smaller than one. These spaces occur naturally in approximation theory…

There has been a long-standing conjecture in Banach algebra that every amenable operator is similar to a normal operator. In this paper, we study the structure of amenable operators on Hilbert spaces. At first, we show that the conjecture…

This paper is a sequel to [6]. In that paper we transferred the discussions in [1] and [13] concerning almost invariant half-spaces for operators on complex Banach spaces to the context of operators on Hilbert space, and we gave easier…

The duality of uniform approximation property for Banach spaces is well known. In this note, we establish, under the assumption of local reflexivity, the duality of uniform approximation property in the category of operator spaces.

Let $T$ be a bounded linear operator on a (real or complex) Banach space $X$. If $(a_n)$ is a sequence of non-negative numbers tending to 0. Then, the set of $x \in X$ such that $\|T^nx\| \geqslant a_n \|T^n\|$ for infinitely many $n$'s has…

Let X be a real Banach space. We prove that the existence of an injective, positive, symmetric and not strictly singular operator from X into its dual implies that either X admits an equivalent Hilbertian norm or it contains a nontrivially…

A necessary and sufficient condition for an operator space to support a multiplication making it completely isometric and isomorphic to a unital operator algebra is proved. The condition involves only the holomorphic structure of the Banach…

We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator R : X --> X such that the set A = {x in X : ||R^n(x)|| --> infinity} is non-empty and nowhere dense in X. Moreover, if…

We show that there is an operator space notion of Lipschitz embeddability between operator spaces which is strictly weaker than its linear counterpart but which is still strong enough to impose linear restrictions on operator space…

Given a separable Banach space $E$, we construct an extremely non-complex Banach space (i.e. a space satisfying that $\|Id + T^2\|=1+\|T^2\|$ for every bounded linear operator $T$ on it) whose dual contains $E^*$ as an $L$-summand. We also…

We consider similarity transformations of a perturbed linear operator $A-B$ in a complex Banach space $\mathcal{X}$, where the unperturbed operator $A$ is a generator of a Banach $L_1(\mathbb{R})$-module and the perturbation operator $B$ is…