Mathematics

We develop a quantum harmonic analysis framework for the affine group. This encapsulates several examples in the literature such as affine localization operators, covariant integral quantizations, and affine quadratic time-frequency representations. In the process, we develop a notion of admissibility for operators and extend well known results to the operator setting. A major theme of the paper is the interaction between operator convolutions, affine Weyl quantization, and admissibility.

This is an expository paper on tensor products where the standard approaches for constructing concrete instances of algebraic tensor products of linear spaces, via quotient spaces or via linear maps of bilinear maps, are reviewed by reducing them to different but isomorphic interpretations of an abstract notion, viz., the universal property, which is based on a pair of axioms.

When improving results about generalized inverses, the aim often is to do this in the most general setting possible by eliminating superfluous assumptions and by simplifying some of the conditions in statements. In this paper, we use Hartwig's well-known triple reverse order law as an example for showing how this can be done using a recent framework for algebraic proofs and the software package OperatorGB. Our improvements of Hartwig's result are proven in rings with involution and we discuss computer-assisted proofs that show these results in other settings based on the framework and a single computation with noncommutative polynomials.

Real-life problems are governed by equations which are nonlinear in nature. Nonlinear equations occur in modeling problems, such as minimizing costs in industries and minimizing risks in businesses. A technique which does not involve the assumption of existence of a real constant whose calculation is unclear is used to obtain a strong convergence result for nonlinear equations of (p, {\eta})-strongly monotone type, where {\eta} > 0, p > 1. An example is presented for the nonlinear equations of (p, {\eta})-strongly monotone type. As a consequence of the main result, the solutions of convex minimization and variational inequality problems are obtained. This solution has applications in other fields such as engineering, physics, biology, chemistry, economics, and game theory.

Let F be an ordered topological vector space (over R ) whose positive cone F + is weakly closed, and let E⊆F be a subspace. We prove that the set of positive continuous linear functionals on E that can be extended (positively and continuously) to F is weak- ∗ dense in the topological dual wedge E ′ + . Furthermore, we show that this result cannot be generalized to arbitrary positive operators, even in finite-dimensional spaces.

Let Ω be an open subset of R N , and let p,q:Ω?�[1,?�] be measurable functions. We give a necessary and sufficient condition for the embedding of the variable exponent space L p(?? (Ω) in L q(?? (Ω) to be almost compact. This leads to a condition on Ω,p and q sufficient to ensure that the Sobolev space W 1,p(?? (Ω) based on L p(?? (Ω) is compactly embedded in L q(?? (Ω); compact embedding results of this type already in the literature are included as special cases.

We study distinguished objects in the category BL of Banach lattices and lattice homomorphisms. The free Banach lattice construction introduced by de Pagter and Wickstead generates push-outs, and combining this with an old result of Kellerer on marginal measures, the amalgamation property of Banach lattices is established. This will be the key tool to prove that L 1 ([0,1 ] c ) is separably BL -injective, as well as to give more abstract examples of Banach lattices of universal disposition for separable sublattices. Finally, an analysis of the ideals on C(Δ, L 1 ) , which is a separably universal Banach lattice as shown by Leung, Li, Oikhberg and Tursi, allows us to conclude that separably BL -injective Banach lattices are necessarily non-separable.

We construct an Asplund Banach space X with a norming Markuševič basis such that X is not weakly compactly generated. This solves a long-standing open problem from the early nineties, originally due to Gilles Godefroy. En route to the proof, we construct a peculiar example of scattered compact space, that also solves a question due to Wiesław Kubiś and Arkady Leiderman.

Pisier's inequality is central in the study of normed spaces and has important applications in geometry. We provide an elementary proof of this inequality, which avoids some non-constructive steps from previous proofs. Our goal is to make the inequality and its proof more accessible, because we think they will find additional applications. We demonstrate this with a new type of restriction on the Fourier spectrum of bounded functions on the discrete cube.

We prove a theorem, which generalises C. Franchetti's estimate for the norm of a projection onto a rich subspace of L p ([0,1]) and the authors' related estimate for compact operators on L p ([0,1]) , 1≤p<∞ .