Banach Actions Preserving Unconditional Convergence

Let A,X,Y be Banach spaces and A×X→Y, (a,x)↦ax be a continuous bilinear function, called a Banach action. We say that this action preserves unconditional convergence if for every bounded sequence (an)n∈ω in A and unconditionally convergent series ∑n∈ωxn in X, the series ∑n∈ωanxn is unconditionally convergent in Y. We prove that a Banach action A×X→Y preserves unconditional convergence if and only if for any linear functional y*∈Y* the operator Dy*:X→A*, Dy*(x)(a)=y*(ax) is absolutely summing. Combining this characterization with the famous Grothendieck theorem on the absolute summability of operators from ℓ1 to ℓ2, we prove that a Banach action A×X→Y preserves unconditional convergence if A is a Hilbert space possessing an orthonormal basis (en)n∈ω such that for every x∈X, the series ∑n∈ωenx is weakly absolutely convergent. Applying known results of Garling on the absolute summability of diagonal operators between sequence spaces, we prove that for (finite or infinite) numbers p,q,r∈[1,∞] with 1r≤1p+1q, the coordinatewise multiplication ℓp×ℓq→ℓr preserves unconditional convergence if and only if one of the following conditions holds: (i) p≤2 and q≤r, (ii) 2<p<q≤r, (iii) 2<p=q<r, (iv) r=∞, (v) 2≤q<p≤r, (vi) q<2<p and 1p+1q≥1r+12.

On the Convergence of Orbits in Sequence Space l^2

This paper discusses the convergence of orbits for diagonal operators defined on . In particular, the basis elements of are obtained using the linear combinations of the elements of the orbit. Furthermore, via the classical result of the determinant of the Vandermonde matrix, it is shown that, the more the elements of the orbit are used, the faster the convergence of the orbit to the basis elements of . Keywords: Diagonal operators; Convergence of Orbits of operators; Vandermonde matrix; Norm topology

A non-diagonalizable pure state

We construct a pure state on the C*-algebra B(ℓ2) of all bounded linear operators on ℓ2, which is not diagonalizable [i.e., it is not of the form limu⟨T(ek),ek⟩ for any orthonormal basis (ek)k∈N of ℓ2 and an ultrafilter u on N]. This constitutes a counterexample to Anderson’s conjecture without additional hypothesis and improves results of C. Akemann, N. Weaver, I. Farah, and I. Smythe who constructed such states making additional set-theoretic assumptions. It follows from results of J. Anderson and the positive solution to the Kadison–Singer problem due to A. Marcus, D. Spielman, and N. Srivastava that the restriction of our pure state to any atomic masa D((ek)k∈N) of diagonal operators with respect to an orthonormal basis (ek)k∈N is not multiplicative on D((ek)k∈N).

Nuclear Embeddings in Weighted Function Spaces

We study nuclear embeddings for weighted spaces of Besov and Triebel–Lizorkin type where the weight belongs to some Muckenhoupt class and is essentially of polynomial type. Here we can extend our previous results concerning the compactness of corresponding embeddings. The concept of nuclearity was introduced by A. Grothendieck in 1955. Recently there is a refreshed interest to study such questions. This led us to the investigation in the weighted setting. We obtain complete characterisations for the nuclearity of the corresponding embedding. Essential tools are a discretisation in terms of wavelet bases, operator ideal techniques, as well as a very useful result of Tong about the nuclearity of diagonal operators acting in $$\ell _p$$ ℓ p spaces. In that way we can further contribute to the characterisation of nuclear embeddings of function spaces on domains.