Related papers: Remarks on strictly singular operators

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…

Let $H_1$ and $H_2$ be complex Hilbert spaces and $T:H_1\rightarrow H_2$ be a bounded linear operator. We say $T$ to be norm attaining, if there exists $x\in H_1$ with $\|x\|=1$ such that $\|Tx\|=\|T\|$. If for every closed subspace $M$ of…

In this article we consider the classical singular integral operator over a local field with rough kernels. We study the boundedness of such an operator on different function spaces by relaxing the smoothness condition on kernels.

Let $X$ be a complex Banach space and let $T$ be a bounded linear operator on $X$. For any closed $T$-invariant subspace $F$ of $X$, $T$ induces operators $T_{|F}:F \longrightarrow F$ and $T/F:X/F\longrightarrow X/F$. In this note, we give…

A differential operator $T$ satisfies the $L^2$-unique continuation property if every $L^2$-solution of $T$ that vanishes on an open subset vanishes identically. We study the $L^2$-unique continuation property of an operator $T$ acting on a…

We shall say that a densely defined closed operator $T$ on a Hilbert space is balanced if $\cD(T)=\cD(T^*)$. Balanced operators are described in terms of their phase operators abnd their moduli. Examples of balanced operators are developed.…

Let $X$ and $Y$ be separable Banach spaces and $T:X\to Y$ be a bounded linear operator. We characterize the non-separability of $T^*(Y^*)$ by means of fixing properties of the operator $T$.

In this article we investigate the disjointly non-singular (DNS) operators. Following [8] we say that an operator $T$ from a Banach lattice $F$ into a Banach space $E$ is DNS, if no restriction of $T$ to a subspace generated by a disjoint…

In this article, we give a representation of bounded complex linear operators which preserve idempotent elements on the Fourier algebra of a locally compact group. When such an operator is moreover positive or contractive, we show that the…

We continue our analysis of the consequences of the commutation relation $[S,T]=\Id$, where $S$ and $T$ are two closable unbounded operators. The {\em weak} sense of this commutator is given in terms of the inner product of the Hilbert…

We study a hypercyclicity property of linear dynamical systems: a bounded linear operator T acting on a separable infinite-dimensional Banach space X is said to be hypercyclic if there exists a vector x in X such that {T^{n}x : n>0} is…

The primary purpose of this paper is to investigate the question of invertibility of the sum of operators. The setting is bounded and unbounded linear operators. Some interesting examples and consequences are given. As an illustrative…

We construct a strictly singular non-compact operator on Gowers' and Maurey's space $GM$.

It is proved that for every stratifiable space $Y$ and a closed subset $X\subset Y$ there exists a regular (i.e. linear positive with unit norm) extension operator $T:C(X\times X)\to C(Y\times Y)$ preserving the class of (pseudo)metrics.…

Four possible definitions of the commutation relation $[S,T]=\Id$ of two closable unbounded operators $S,T$ are compared. The {\em weak} sense of this commutator is given in terms of the inner product of the Hilbert space $\H$ where the…

We prove that a bounded linear operator $T$ is a direct sum of an invertible operator and an operator with at most countable spectrum iff $0\notin\mbox{acc}^{\omega_{1}}\,\sigma(T),$ where $\omega_{1}$ is the smallest uncountable ordinal…

We consider a class of bounded linear operators between Banach spaces, which we call operators with the Kato property, that includes the family of strictly singular operators between those spaces. We show that if $T:E\to F$ is a dense-range…

We consider linear spectral-meromorphic (s-meromorphic) OD operators at the real axis such that all local solutions to the eigenvalue problems are meromorphic for all $\lambda$. By definition, rank one algebro-geometrical operator $L$ admit…

A bounded linear operator $T$ on a Banach space $X$ is called hypercyclic if there exists a vector $x \in X$ such that $orb{(x,T)}$ is dense in $X$. The Hypercyclicity Criterion is a well-known sufficient condition for an operator to be…

In this paper, we introduce the B-discrete spectrum of an unbounded closed operator and we prove that a closed operator has a purely B-discrete spectrum if and only if it has a meromorphic resolvent. After that, we study the stability of…