In this short note, we discuss essential positivity of Toeplitz operators on the Fock space, as motivated by a recent question of Perälä and Virtanen (Proc. Amer. Math. Soc. 151:4807–4815, 2023). We give a proper characterization of essential positivity in terms of limit operators. A conjectured characterization of essential positivity of Perälä and Virtanen is disproven when the assumption of radiality is dropped. Nevertheless, when the symbol of the Toeplitz operator is of vanishing mean oscillation, we show that the conjecture of Perälä and Virtanen holds true, even without radiality.

In this article, we study a commutative Banach algebra structure on the space L1(R2n)⊕T1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1(\\mathbb {R}^{2n})\\oplus {\\mathcal {T}}^1$$\\end{document}, where the T1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}^1$$\\end{document} denotes the trace class operators on L2(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(\\mathbb {R}^{n})$$\\end{document}. The product of this space is given by the convolutions in quantum harmonic analysis. Towards this goal, we study the closed ideals of this space, and in particular its Gelfand theory. We additionally develop the concept of quantum Segal algebras as an analogue of Segal algebras. We prove that many of the properties of Segal algebras have transfers to quantum Segal algebras. However, it should be noted that in contrast to Segal algebras, quantum Segal algebras are not ideals of the ambient space. We also give examples of different constructions that yield quantum Segal algebras.

We study numerical ranges of antilinear operators on both Hilbert and Banach spaces. We prove that the numerical range of an antilinear operator on at least two-dimensional space is always a disc, and so a convex set. This improves existing results. We also study other properties known for numerical ranges of linear operators and discuss similarities and differences in the antilinear setting.

A unital C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebra is called N-subhomogeneous if its irreducible representations are finite dimensional with dimension at most N. We extend this notion to operator systems, replacing irreducible representations by boundary representations. This is done by considering UCP(S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {UCP }(\\mathcal {S})$$\\end{document} which is the matrix state space associated with an operator system S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {S}$$\\end{document} and identifying the boundary representations as absolute matrix extreme points. We show that two N-subhomogeneous operator systems are completely order equivalent if and only if they are N-order equivalent. Moreover, we show that a unital N-positive map into a finite dimensional N-subhomogeneous operator system is completely positive. We apply these tools to classify pairs of q-commuting unitaries up to ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}-isomorphism. Similar results are obtained for operator systems related to higher dimensional non-commutative tori.

AbstractWe study abstract sufficient criteria for cost-uniform open-loop stabilizability of linear control systems in a Banach space with a bounded control operator, which build up and generalize a sufficient condition for null-controllability in Banach spaces given by an uncertainty principle and a dissipation estimate. For stabilizability these estimates are only needed for a single spectral parameter and, in particular, their constants do not depend on the growth rate w.r.t. this parameter. Our result unifies and generalizes earlier results obtained in the context of Hilbert spaces. As an application we consider fractional powers of elliptic differential operators with constant coefficients in $$L_p(\mathbb {R}^d)$$ L p ( R d ) for $$p\in [1,\infty )$$ p ∈ [ 1 , ∞ ) and thick control sets.

We study the gauge-invariant ideal structure of the Nica-Toeplitz algebra NT(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}\\mathcal {T}(X)$$\\end{document} of a product system (A, X) over Nn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {N}^n$$\\end{document}. We obtain a clear description of X-invariant ideals in A, that is, restrictions of gauge-invariant ideals in NT(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N\\hspace{-1.111pt}T}(X)$$\\end{document} to A. The main result is a classification of gauge-invariant ideals in NT(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N\\hspace{-1.111pt}T}(X)$$\\end{document} for a proper product system in terms of families of ideals in A. We also apply our results to higher-rank graphs.