It is well known that the only proper non-trivial norm closed ideal in the algebra L(X) for $X={\ell }_p$ (1 ≤ p < ∞) or X = c₀ is the ideal of compact operators. The next natural question is to describe all closed ideals of $L\left({\ell }_p\oplus {\ell }_q\right)$ for 1 ≤ p,q < ∞, p ≠ q, or equivalently, the closed ideals in $L({\ell }_p,{\ell }_q)$ for p < q. This paper shows that for 1 < p < 2 < q < ∞ there are at least four distinct proper closed ideals in $L({\ell }_p,{\ell }_q)$, including one that has not been studied before. The proofs use various methods from Banach...

In the paper the geometric properties of the positive cone and positive part of the unit ball of the space of operator-valued continuous space are discussed. In particular we show that $\text{ext-ray}{\text{C}}_{+}(K,ℒ\left(H\right))=\{{ℝ}_{+}{\mathbf{1}}_{\left\{k_0\right\}}𝐱\otimes 𝐱:𝐱\in 𝐒\left(H\right),k_0\text{is}\text{an}\text{isolated}\text{point}\text{of}K\}$$\text{ext}{𝐁}_{+}...$

The paper introduces a notion of quasi-compact operator net on a Banach space. It is proved that quasi-compactness of a uniform Lotz-Räbiger net ${\left(T_{\lambda }\right)}_{\lambda }$ is equivalent to quasi-compactness of some operator $T_{\lambda }$. We prove that strong convergence of a quasi-compact uniform Lotz-Räbiger net implies uniform convergence to a finite-rank projection. Precompactness of operator nets is also investigated.

In this paper we study the reflexive subobject lattices and reflexive endomorphism algebras in a concrete category. For the category Set of sets and mappings, a complete characterization for both reflexive subobject lattices and reflexive endomorphism algebras is obtained. Some partial results are also proved for the category of abelian groups.

In normed linear space settings, modifying the sequential definition of continuity of an operator by replacing the usual limit "$lim$" with arbitrary linear regular summability methods $𝐆$ we consider the notion of a generalized continuity ($({𝐆}_{\mathbf{1}},{𝐆}_{\mathbf{2}})$-continuity) and examine some of its consequences in respect of usual continuity and linearity of the operators between two normed linear spaces.

If G is a discrete group, the algebra CD(G) of convolution dominated operators on l²(G) (see Definition 1 below) is canonically isomorphic to a twisted L¹-algebra $l¹(G,l^{\infty }\left(G\right),T)$. For amenable and rigidly symmetric G we use this to show that any element of this algebra is invertible in the algebra itself if and only if it is invertible as a bounded operator on l²(G), i.e. CD(G) is spectral in the algebra of all bounded operators. For G commutative, this result is known (see [1], [6]), for G noncommutative discrete...

The algebra B(ℋ) of all bounded operators on a Hilbert space ℋ is generated in the strong operator topology by a single one-dimensional projection and a family of commuting unitary operators with cardinality not exceeding dim ℋ. This answers Problem 8 posed by W. Żelazko in [6].

We introduce and study a new concept of strongly $l_p$-summing m-linear operators in the category of operator spaces. We give some characterizations of this notion such as the Pietsch domination theorem and we show that an m-linear operator is strongly $l_p$-summing if and only if its adjoint is $l_p$-summing.

We show that a Banach space X has the compact approximation property if and only if for every Banach space Y and every weakly compact operator T: Y → X, the space = S ∘ T: S compact operator on X is an ideal in = span(,T) if and only if for every Banach space Y and every weakly compact operator T: Y → X, there is a net $\left(S_{\gamma }\right)$ of compact operators on X such that $su{p}_{\gamma }\left|\right|S_{\gamma }T\left|\right|\le \left|\right|T\left|\right|$ and $S_{\gamma }\to I_X$ in the strong operator topology. Similar results for dual spaces are also proved.