For a pair $$(A,B)$$ of not necessarily bounded and commuting self-adjoint operators for function f on the Euclidean space $${{\mathbb{R}}^{2}}$$ that belongs to inhomogeneous Besov class $$B_{{\infty ,1}}^{1}({{\mathbb{R}}^{2}})$$ , we define $$f(A,B)$$ these as densely defined operator. We consider problem estimating functions under perturbations . It is established if $$1 \leqslant p 2$$ $$(...

Let Πn denote the set of all set partitions of {1, 2, . . . , n}. We consider two subsets of Πn, one connected to rook theory and one associated with symmetric functions in noncommuting variables. Let En ⊆ Πn be the subset of all partitions corresponding to an extendable rook (placement) on the upper-triangular board, Tn−1. Given π ∈ Πm and σ ∈ Πn, define their slash product to be π|σ = π∪(σ+m)...

We examine the problem of defining Lagrangian and Hamiltonian mechanics for a particle moving on a quantum plane Qq,p. For Lagrangian mechanics, we first define a tangent quantum plane TQq,p spanned by noncommuting particle coordinates and velocities. Using techniques similar to those of Wess and Zumino, we construct two different differential calculi on TQq,p. These two differential calculi ca...

The study of matrix inequalities in a dimension-free setting is in the realm of free real algebraic geometry (RAG). In this paper we investigate constrained trace and eigenvalue optimization of noncommutative polynomials. We present Lasserre’s relaxation scheme for trace optimization based on semidefinite programming (SDP) and demonstrate its convergence properties. Finite convergence of this r...

We formally prove the ultimate statement: Local realistic theories for quantum measurements could not impose the commutative algebraic structure into the set of all quantum observables even if all quantum statistics (including every quantum state and every observable) could be reproduced by local realistic theories. Thus, it turns out that the result, using a pair of projectors, by Malley et al...

Quantum Euclidean spaces, as Moyal deformations of are the model examples noncompact noncommutative manifold. In this paper, we study quantum space equipped with partial derivatives satisfying canonical commutation relation (CCR). This gives an example semifinite spectral triple nonflat geometric structure. We develop abstract symbol calculus for pseudo-differential operators noncommuting deriv...

Werner states are multipartite quantum that invariant under the diagonal conjugate action of unitary group. This paper gives a complete characterization their entanglement is independent underlying local Hilbert space: for every entangled state there exists dimension-free witness. The construction such witness formulated as an optimization problem. To solve it, two semidefinite programming hier...