Let X be a closed subspace of a Banach space W and let F be the operator ideal of finite-rank operators. If α is a tensor norm, A is a Banach operator ideal, and λ > 0, then we call the condition “‖S‖α ≤ λ‖S‖A(X,W ) for all S ∈ F(X,X)” an inner inequality and the condition “‖T‖α ≤ λ‖T‖A(Y,W ) for all Banach spaces Y and for all T ∈ F(Y,X)” an outer inequality. We describe cases when outer inequ...

In this note, we aim to present some properties of the space of all weakly fuzzy bounded linear operators, with the Bag and Samanta’s operator norm on Felbin’s-type fuzzy normed spaces. In particular, the completeness of this space is studied. By some counterexamples, it is shown that the inverse mapping theorem and the Banach-Steinhaus’s theorem, are not valid for this fuzzy setting. Also...

The classical Hahn-Banach Theorem states that any linear bounded functional defined on a linear subspace of a normed space admits a norm-preserving linear bounded extension to the whole space. The constructive and computational content of this theorem has been studied by Bishop, Bridges, Metakides, Nerode, Shore, Kalantari, Downey, Ishihara and others and it is known that the theorem does not a...

Let A be a C∗-algebra, and let X be a Banach A-bimodule. B. E. Johnson showed that local derivations from A into X are derivations. We extend this concept of locality to the higher cohomology of a C ∗-algebra and show that, for every n ∈ N, bounded local n-cocycles from A into X are n-cocycles. The study of the local properties of Hochschild cohomology of a Banach algebra was initiated by intro...

During the last years the dynamics of linear operators on infinite dimensional spaces has been extensively studied, see the survey articles [3], [6], [7], [8], [9], [11]. Let us recall the notion of hypercyclicity. Let X be a separable Banach space and T : X → X be a bounded linear operator. The operator T is said to be hypercyclic provided there exists a vector x ∈ X such that its orbit under ...

In recent years, Fourier multiplier theorems in vector–valued function spaces have found many applications in embedding theorems of abstract function spaces and in theory of differential operator equations, especially in maximal regularity of parabolic and elliptic differential–operator equations. Operator–valued multiplier theorems in Banach–valued function spaces have been discussed extensive...

We show that if X is a L∞-space with the Dieudonnè property and Y is a Banach space not containing l1, then any operator T : X⊗ Y → Z, where Z is a weakly sequentially complete Banach space, is weakly compact. Some other results of the same kind are obtained. Let X be a L∞-space (see [1] for this notion and some useful results on L∞-spaces) and Y be a Banach space not containing l1.We consider ...

In addition to Pisier’s counterexample of a non-accessible maximal Banach ideal, we will give a large class of maximal Banach ideals which are accessible. The first step is implied by the observation that a ”good behaviour” of trace duality, which is canonically induced by conjugate operator ideals can be extended to adjoint Banach ideals, if and only if these adjoint ideals satisfy an accessib...