Related papers: Remarks on bounded operators in $\ell$-K\"othe spa…

Let $X, Y$ be complete metric spaces and $E, F$ be Banach spaces. A bijective linear operator from a space of $E$-valued functions on $X$ to a space of $F$-valued functions on $Y$ is said to be biseparating if $f$ and $g$ are disjoint if…

Well-bounded operators are linear operators on a Banach space $X$ that have an $AC[a,b]$ functional calculus for some interval $[a,b]$. A well-bounded operator is of type (B) if it can be written as an integral against a spectral family of…

Every closed subspace of each of the Banach spaces $X = \ell_p(\Gamma)$ and $X=c_0(\Gamma)$, where $\Gamma$ is a set and $1<p<\infty$, is the kernel of a bounded operator $X\to X$. On the other hand, whenever $\Gamma$ is an uncountable set,…

In the short note we prove that for every $0<p<1$, there exists an infinite dimensional closed linear subspace of $\mathcal{L}\left( \ell_{p};\ell_{p}\right) $ every nonzero element of which is non $(r,s)$-absolutely summing operator for…

We study left symmetric and right symmetric elements in the space $\ell_{\infty}(K, \mathbb{X}) $ of bounded functions from a non-empty set $K$ to a Banach space $\mathbb{X}.$ We prove that a non-zero element $ f \in\ell_{\infty}(K,…

Let $E$ be a Banach space and $\X$ be the closed unit ball of the dual space $E^*$. For a compact set $K$ in $E$, we prove that $K$ is polynomially convex in $E$ if and only if there exist a unital commutative Banach algebra $A$ and a…

We give sufficient conditions on a Banach space $X$ which ensure that $\ell_{\infty}$ embeds in $\mathcal{L}(X)$, the space of all operators on $X$. We say that a basic sequence $(e_n)$ is quasisubsymmetric if for any two increasing…

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 $T$ 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…

Given an operator $\phi:X\rightarrow Y$ between Banach spaces, we consider its tensor powers $\phi^{\otimes k}$ as operators from the $k$-fold injective tensor product of $X$ to the $k$-fold projective tensor product of $Y$. We show that…

A banach space X is a normed vector space, which is complete with respect to the metric induced by the norm. Given a bounded linear operator T acting on a banach space X, T is said to attain its norm if there is a unit vector z in X, such…

Consider two continuous linear operators $T\colon X_1(\mu)\to Y_1(\nu)$ and $S\colon X_2(\mu)\to Y_2(\nu)$ between Banach function spaces related to different $\sigma$-finite measures $\mu$ and $\nu$. We characterize by means of weighted…

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…

It is well-known that in Banach spaces with finite cotype, the $R$-bounded and $\gamma$-bounded families of operators coincide. If in addition $X$ is a Banach lattice, then these notions can be expressed as square function estimates. It is…

In this paper we introduce the notion of property $(BR)$ and property $(BgR)$ for bounded linear operators defined on an infinite-dimensional Banach space. These properties in connection with Weyl type theorems and in the frame of polaroid…

We characterize $k-$smoothness of an element on the unit sphere of a finite-dimensional polyhedral Banach space. Then we study $k-$smoothness of an operator $T \in \mathbb{L}(\ell_{\infty}^n,\mathbb{Y}),$ where $\mathbb{Y}$ is a…

In this paper we introduce the strong Bishop-Phelps-Bollob\'as property (sBPBp) for bounded linear operators between two Banach spaces $X$ and $Y$. This property is motivated by a Kim-Lee result which states, under our notation, that a…

Given an infinite matrix $M=(m_{nk})$ we study a family of sequence spaces $\ell_M^p$ associated with it. When equipped with a suitable norm $\|\cdot\|_{M,p}$ we prove some basic properties of the Banach spaces of sequences…

We study the class of Banach spaces $X$ such that the locally convex space $(X,\mu(X,Y))$ is complete for every norming and norm-closed subspace $Y \subset X^*$, where $\mu(X,Y)$ denotes the Mackey topology on $X$ associated to the dual…

It is proved that: each collectively order continuous set of operators from an Archimedean OVS with a generating cone to an OVS is collectively order bounded; and each collectively order to norm bounded set of operators from an ordered…