We show that the Connes-Rovelli thermal time associated with the quantum harmonic oscillator can be described as an (unsharp) observable, that is, as a positive operator valued measure. We furthermore present extensions of this result to the free massless relativistic particle in one dimension and to a hypothetical physical system whose equilibrium state is given by the noncommutative integral.

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We consider operators acting on a UMD Banach lattice X that have the same algebraic structure as the position and momentum operators associated with the harmonic oscillator (Formula Presented) acting on L²(R^d). More precisely, we consider abstract harmonic oscillators of the form (Formula Presented) for tuples of operators (Formula Presented), where i A_j and iB_k are assumed to generate C₀ groups and to satisfy the canonical commutator relations. We prove functional calculus results for these abstract harmonic oscillators that match classical Hörmander spectral multiplier estimates for the harmonic oscillator (Formula Presented). This covers situations where the underlying metric measure space is not doubling and the use of function spaces that are not particularly well suited to extrapolation arguments. For instance, as an application we treat the harmonic oscillator on mixed norm Bargmann–Fock spaces. Our approach is based on a transference principle for the Schrödinger representation of the Heisenberg group that allows us to reduce the problem to the study of the twisted Laplacian on the Bochner spaces L²(R^2d; X). This can be seen as a generalisation of the Stone–von Neumann theorem to UMD lattices X that are not Hilbert spaces.

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We generalise the classical Weyl pseudo-differential calculus on R^d to the setting of two d-tuples of operators A = (A₁,..., A_d) and B = (B₁,..., B_d) acting on a Banach space generating bounded C0-groups satisfying the Weyl canonical commutation relations. We show that the resulting Weyl calculus extends to symbols in the standard symbol class S⁰ provided appropriate bounds can be established. Using transference techniques we obtain boundedness of the H^∞-functional calculus (and even the Hormander calculus), for the abstract harmonic oscillator.

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We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of time-dependent semilinear parabolic problems with noise given by a cylindrical Brownian motion. We treat random coefficients that are only progressively measurable in the time variable. For 2m-th order systems with VMO regularity in space, we obtain L ^p (L ^q ) estimates for all p> 2 and q≥ 2 , leading to optimal space-time regularity results. For second order systems with continuous coefficients in space, we also include a first order linear term, under a stochastic parabolicity condition, and obtain L ^p (L ^p ) estimates together with optimal space-time regularity. For linear second order equations in divergence form with random coefficients that are merely measurable in both space and time, we obtain estimates in the tent spaces Tσp,2 of Coifman–Meyer–Stein. This is done in the deterministic case under no extra assumption, and in the stochastic case under the assumption that the coefficients are divergence free.

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Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as investigated in the papers by Axelsson, Keith and the second author, provided insight into the solution of the Kato square-root problem for elliptic operators in L² spaces and allowed for an extension of these estimates to other systems with applications to non-smooth boundary value problems. In this paper, we determine conditions under which such operators satisfy conical square function estimates in a range of L^p spaces, thus allowing us to apply the theory of Hardy spaces associated with an operator to prove that they have a bounded holomorphic functional calculus in those L^p spaces. We also obtain functional calculus results for restrictions to certain subspaces, for a larger range of p. This provides a framework for obtaining L^p results on perturbed Hodge Laplacians, generalising known Riesz transform bounds for an elliptic operator L with bounded measurable coefficients, one Sobolev exponent below the Hodge exponent, and L^p bounds on the square-root of L by the gradient, two Sobolev exponents below the Hodge exponent. Our proof shows that the heart of the harmonic analysis in L² extends to L^p for all p ∈ (1,∞), while the restrictions in p come from the operator-theoretic part of the L² proof. In the course of our work, we obtain some results of independent interest about singular integral operators on tent spaces and about the relationship between conical and vertical square functions.

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In this paper, we introduce a Weyl functional calculus a↦a(Q,P) for the position and momentum operators Q and P associated with the Ornstein-Uhlenbeck operator L=−Δ+x⋅∇, and give a simple criterion for restricted Lp-Lq boundedness of operators in this functional calculus. The analysis of this non-commutative functional calculus is simpler than the analysis of the functional calculus of~L. It allows us to recover, unify, and extend old and new results concerning the boundedness of exp(−zL) as an operator from Lp(Rd,γα) to Lq(Rd,γβ) for suitable values of z∈C with Rez>0, p,q∈[1,∞), and α,β>0. Here, γτ denotes the centered Gaussian measure on Rd with density (2πτ)−d/2exp(−|x|2/2τ)@en