Abstract In this work we investigate special aspects of positivity preservers and especially diagonal positivity preservers, i.e., linear maps $$T:\mathbb {R}[x_1,\dots ,x_n]\rightarrow \mathbb {R}[x_1,\dots ,x_n]$$ T : R [ x 1 , ⋯ , x n ] → R [ x 1 , ⋯ , x n ] such that $$Tx^\alpha = t_\alpha x^\alpha $$ T x α = t α x α for all $$\alpha \in \mathbb {N}_0^n$$ α ∈ N 0 n with $$t_\alpha \in \mathbb {R}$$ t α ∈ R and $$Tp\ge 0$$ T p ≥ 0 on $$\mathbb {R}^n$$ R n for all $$p\in \mathbb {R}[x_1,\dots ,x_n]$$ p ∈ R [ x 1 , ⋯ , x n ] with $$p\ge 0$$ p ≥ 0 on $$\mathbb {R}^n$$ R n . We discuss representations of T , give characterizations of diagonal positivity preservers, and compare these to previous (partial) results in the literature. On the side we get a characterization of linear maps preserving moment sequences and a new proof of Schur’s product formula. The tool of diagonal positivity preservers simplifies several other existing proofs in the literature. We give a full characterization of generators A of diagonal positivity preservers, i.e., $$e^{tA}$$ e tA is a diagonal positivity preserver for all $$t\ge 0$$ t ≥ 0 . We give the connection of these generators to infinitely divisible moment sequences.

Abstract We establish left and right canonical factorizations of Hilbert-space operator-valued functions G ( z ) that are analytic on neighborhoods of the complex unit circle $${\mathbb {T}}$$ T and the origin 0 and that have the form $$G(z)=I+F(z)$$ G ( z ) = I + F ( z ) with F ( z ) taking strictly contractive values on $${\mathbb {T}}$$ T . Such functions can be realized as transfer functions of infinite dimensional dichotomous discrete-time linear systems, and we employ the strict bounded real lemma for this class of operators, together with associated Kreĭn space theory, to derive explicit formulas for the left and right canonical factorizations.

Abstract We study the Stokes operator with Hodge, Navier, and Robin boundary conditions on domains $$\Omega \subseteq \mathbb {R}^d$$ Ω ⊆ R d that are uniformly $$C^{2,1}$$ C 2 , 1 . Starting with the Hodge Laplacian we etablish a bounded Hörmander functional calculus for the Stokes operator with Hodge boundary conditions. This entails a Hörmander functional calculus and boundedness of the $$H^\infty $$ H ∞ -calculus in spaces of soleniodal vector fields for the Stokes operator with Hodge boundary conditions. We then establish boundedness of the $$H^\infty $$ H ∞ -calculus for Stokes operators with Navier type conditions via Robin type perturbations of Hodge boundary conditions. This implies maximal $$L^p$$ L p -regularity for these operators and results on fractional domain spaces. Our results cover certain non-Helmholtz domains.

We 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 Lp(Rd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p(\\mathbb {R}^d)$$\\end{document} for p∈[1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in [1,\\infty )$$\\end{document} and thick control sets.