The Drury–Arveson Space on the Siegel Upper Half-space and a von Neumann Type Inequality

In this work we study what we call Siegel–dissipative vector of commuting operators $$(A_1,\ldots , A_{d+1})$$ ( A 1 , … , A d + 1 ) on a Hilbert space $${{\mathcal {H}}}$$ H and we obtain a von Neumann type inequality which involves the Drury–Arveson space DA on the Siegel upper half-space $${{\mathcal {U}}}$$ U . The operator $$A_{d+1}$$ A d + 1 is allowed to be unbounded and it is the infinitesimal generator of a contraction semigroup $$\{e^{-i\tau A_{d+1}}\}_{\tau <0}$$ { e - i τ A d + 1 } τ < 0 . We then study the operator $$e^{-i\tau A_{d+1}}A^{\alpha }$$ e - i τ A d + 1 A α where $$A^{\alpha }=A_1^{\alpha _1}\cdots A^{\alpha _d}_d$$ A α = A 1 α 1 ⋯ A d α d for $$\alpha \in {\mathbb N}_0^d$$ α ∈ N 0 d and prove that can be studied by means of model operators on a weighted $$L^2$$ L 2 space. To prove our results we obtain a Paley–Wiener type theorem for DA and we investigate some multiplier operators on DA as well.

Uniform-in-time error estimate of the random batch method for the Cucker–Smale model

We present a uniform-in-time (and in particle numbers as well) error estimate for the random batch method (RBM) [S. Jin, L. Li and J.-G. Liu, Random batch methods (RBM) for interacting particle systems, J. Comput. Phys. 400 (2020) 108877] to the Cucker–Smale (CS) model. The uniform-in-time error estimates of the RBM have been obtained for various interacting particle systems, when corresponding flow generates a contraction semigroup. In this paper, we derive a uniform-in-time error estimate for RBM-approximation to the CS model in which the corresponding flow does not generate contractive semigroup. To derive uniform error estimate, we use asymptotic flocking estimate of the RBM-approximated CS model which yields the decay of relative velocities to zero, at least in the order of [Formula: see text], while velocities of the original system decay exponentially. Here, [Formula: see text] is the decay rate of the communication weight with respect to the distance between particles in the CS model. We also provide several numerical simulations to confirm the analytical results.

Observer Design for Estimation of Nonobservable States in Buildings

Efficient temperature control requires more than air temperature measurements. Relevant variables, such as wall, ceiling, and other construction temperature evolution are usually unmeasured. Estimation of such quantities is often difficult because they are not observable with respect to available data. Their availability however would allow efficient control design. In this paper, we propose a method for designing state observers that efficiently estimate not only observable but also nonobservable (but detectable) state variables. Our method uses contraction semigroup, to obtain observer with a monotonic error reduction. Proposed approach gives twice as fast estimation as pure simulation and avoids transitional error standard observer would have. Problem of state estimation in building control applications is an important one. Attractiveness of obtaining values of physically unmeasurable variables is easily visible, as it would allow more efficient methods of temperature control. In this paper, authors discuss the problem of such estimation using a lumped capacitance model. This type of model is usually only detectable but not observable. Methods of observer tuning for such systems are not discussed properly in the literature and require special consideration. In this paper, three approaches for estimation are compared: pure model, eigenvalue shifting, and contraction semigroup observer. Results are illustrated with numerical experiments.

On the Abstract Subordinated Exit Equation

Let be a -contraction semigroup on a real Banach space . A -exit lawis a -valued function satisfying the functional equation: , . Let be a Bochner subordinator and let be the subordinated semigroup of (in the Bochner sense) by means of . Under some regularity assumption, it is proved in this paper that each -exit law is subordinated to a unique -exit law.

ON PERTURBED POSITIVE SEMIGROUPS ON THE BANACH SPACE OF TRACE CLASS OPERATORS

Let [Formula: see text] be the generator of a positive (i.e. leaving invariant [Formula: see text]) contraction semigroup on [Formula: see text], the space of self-adjoint trace class operators on a Hilbert space H, endowed with the trace norm, and let [Formula: see text] be a positive linear operator such that [Formula: see text], [Formula: see text]. We show that there exists a minimal positive contraction semigroup generated by some [Formula: see text] and provide a systematic study of the total mass carried by individual trajectories [Formula: see text] with non-negative initial data ρ. The analysis relies on mathematical properties of two (a priori) different extensions [Formula: see text] of the functional [Formula: see text] to [Formula: see text].

SOLUTION OF THE MPC PROBLEM IN TERMS OF THE SZ.-NAGY–FOIAŞ DILATION THEORY

Motivated by physical problems, Misra, Prigogine and Courbage (MPC) studied the following problem: given a one-parameter unitary group {Ut} on a separable Hilbert space [Formula: see text], find a Hilbert space [Formula: see text], a contraction semigroup {Wt} on [Formula: see text] and an injective operator [Formula: see text] with dense range which intertwines the actions of {Ut} and {Wt} (ΛWt = Ut Λ). More precisely, they studied the case where [Formula: see text] is an L2-space over a probability space and both {Ut} and {Wt} are Markovian (i.e. positivity and identity preserving). MPC gave a sufficient condition for the existence of a solution of the above problem, the existence of a time operator associated to {Ut}. In this paper we prove that, using the Sz.-Nagy–Foiaş dilation theory, it is possible to give a constructive characterization of all the solutions of the MPC problem in the general context. This criterium allows one to construct a solution of the MPC problem for which no time operator exists. When specialized to L2-spaces and Markovian {Ut} and {Wt}, the present criterium is applied to address the so-called inverse problem of Statistical Mechanics, namely to characterize the intrinsically random dynamics {Ut}.

A strong convergence theorem for contraction semigroups in Banach spaces

We establish a Banach space version of a theorem of Suzuki [8]. More precisely we prove that if X is a uniformly convex Banach space with a weakly continuous duality map (for example, lp for 1 < p < ∞), if C is a closed convex subset of X, and if F = {T (t): t ≥ 0} is a contraction semigroup on C such that Fix(F) ≠ ∅, then under certain appropriate assumptions made on the sequences {αn} and {tn} of the parameters, we show that the sequence {xn} implicitly defined byfor all n ≥ 1 converges strongly to a member of Fix(F).

SECOND QUANTIZATION AND THE Lp-SPECTRUM OF NONSYMMETRIC ORNSTEIN–UHLENBECK OPERATORS

The spectra of the second quantization and the symmetric second quantization of a strict Hilbert space contraction are computed explicitly and shown to coincide. As an application, we compute the spectrum of the nonsymmetric Ornstein–Uhlenbeck operator L associated with the infinite-dimensional Langevin equation [Formula: see text] where A is the generator of a strongly continuous semigroup on a Banach space E and W is a cylindrical Wiener process in E. Assuming the existence of an invariant measure μ for L, under suitable assumptions on A we show that the spectrum of L in the space Lp (E, μ) (1< p< ∞) is given by [Formula: see text] where Aμ is the generator of a Hilbert space contraction semigroup canonically associated with A and μ. We prove that the assumptions on A are always satisfied in the strong Feller case and in the finite-dimensional case. In the latter case we recover the recent Metafune–Pallara–Priola formula for σ(L).