Continuous One-parameter Group Research Articles

A Generalized Polar-Coordinate Integration Formula with Applications to the Study of Convolution Powers of Complex-Valued Functions on $${\mathbb {Z}}^d$$

In this article, we consider a class of functions on $${\mathbb {R}}^d$$ , called positive homogeneous functions, which interact well with certain continuous one-parameter groups of (generally anisotropic) dilations. Generalizing the Euclidean norm, positive homogeneous functions appear naturally in the study of convolution powers of complex-valued functions on $${\mathbb {Z}}^d$$ . As the spherical measure is a Radon measure on the unit sphere which is invariant under the symmetry group of the Euclidean norm, to each positive homogeneous function P, we construct a Radon measure $$\sigma _P$$ on $$S=\{\eta \in {\mathbb {R}}^d:P(\eta )=1\}$$ which is invariant under the symmetry group of P. With this measure, we prove a generalization of the classical polar-coordinate integration formula and deduce a number of corollaries in this setting. We then turn to the study of convolution powers of complex functions on $${\mathbb {Z}}^d$$ and certain oscillatory integrals which arise naturally in that context. Armed with our integration formula and the Van der Corput lemma, we establish sup-norm-type estimates for convolution powers; this result is new and partially extends results of Randles and Saloff-Coste (J Fourier Anal Appl 21(4):754–798, 2015; Rev Mat Iberoam 33(3):1045–1121, 2017).

KMS states on the Toeplitz algebras of higher-rank graphs

The Toeplitz algebra TC⁎(Λ) for a finite k-graph Λ is equipped with a continuous one-parameter group αr for each r∈Rk, obtained by composing the map R∋t→(eitr1,…,eitrk)∈Tk with the gauge action on TC⁎(Λ). In this paper we give a complete description of the β-KMS states for the C⁎-dynamical system (TC⁎(Λ),αr) for all finite k-graphs Λ and all values of β∈R and r∈Rk.

Symmetries of the KMS Simplex

A continuous groupoid homomorphism $c$ on a locally compact second countable Hausdorff \'etale groupoid $\mathcal{G}$ gives rise to a $C^{*}$-dynamical system in which every $\beta$-KMS state can be associated to a $e^{-\beta c}$-quasi-invariant measure $\mu$ on $\mathcal{G}^{(0)}$. Letting $\Delta_{\mu}$ denote the set of KMS states associated to such a $\mu$, we will prove that $\Delta_{\mu}$ is a simplex for a large class of groupoids, and we will show that there is an abelian group that acts transitively and freely on the extremal points of $\Delta_{\mu}$. This group can be described using the support of $\mu$, so our theory of symmetries can be used to obtain a description of all KMS states by describing the $e^{-\beta c}$-quasi-invariant measures. To illustrate this we will describe the KMS states for the Cuntz-Krieger algebras of all finite higher rank graphs without sources and a large class of continuous one-parameter groups.

Representations of a Group of Linear Operators in a Banach Space on the Set of Entire Vectors of its Generator

For a strongly continuous one-parameter group {U(t)} t ∈(−∞,∞) of linear operators in a Banach space $$ \mathfrak{B} $$ with generator A, we prove the existence of a set $$ \mathfrak{B} $$ 1 dense in $$ \mathfrak{B} $$ on the elements x of which the function U(t)x admits an extension to an entire $$ \mathfrak{B} $$ -valued vector function. The description of the vectors from $$ \mathfrak{B} $$ 1 for which this extension has a finite order of growth and a finite type is presented. It is also established that the inclusion x ∈ $$ \mathfrak{B} $$ 1 is a necessary and sufficient condition for the existence of the limit $$ { \lim}_{n\to 1}{\left(I+\frac{tA}{n}\right)}^nx $$ and this limit is equal to U(t)x.

Amenable hyperbolic groups

We give a complete characterization of the locally compact groups that are non-elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting automorphisms. We moreover give a description of all Gromov-hyperbolic locally compact groups with a cocompact amenable subgroup: modulo a compact normal subgroup, these turn out to be either rank one simple Lie groups, or automorphism groups of semi-regular trees acting doubly transitively on the set of ends. As an application, we show that the class of hyperbolic locally compact groups with a cusp-uniform non-uniform lattice, is very restricted.

UNBOUNDED HERMITIAN OPERATORS ON KOLASKI SPACES

AbstractWe investigate strongly continuous one-parameter (C0) groups of isometries acting on certain spaces of analytical functions which were introduced by Kolaski (C. J. Kolaski, Isometries of some smooth normed spaces of analytic functions, Complex Var. Theory Appl. 10(2–3) (1988), 115–122). We characterize the generators of these groups of isometries and also the spectrum of the generators. We provide an example on the Bloch space of an unbounded hermitian operator with non-compact resolvent.

On periodic one-parameter groups of linear operators in a Banach space and applications

Let D be the infinitesimal generator of a strongly continuous periodic one-parameter group of linear operators in a Banach space. Main results: An analog of the resolvent operator (= quasi-resolvent operator of D) is defined for points of the spectrum of D and its evident form is given. The theorem on integral for the operator D, theorems on the existence of periodic solutions of a linear differential equation of the n th order with constant coefficients and systems of linear differential equations with constant coefficients in Banach spaces are obtained. In the case of the existence of periodic solutions, evident forms of all periodic solutions of a linear differential equation of the n th order with constant coefficients and systems of linear differential equations with constant coefficients in Banach spaces are given in terms of resolvent and quasi-resolvent operators of D.MSC:42A, 43, 47D.

GROUP THEORY AND DIFFERENTIAL TRANSFORM ANALYSIS OF MIXED CONVECTIVE HEAT AND MASS TRANSFER FROM A HORIZONTAL SURFACE WITH CHEMICAL REACTION EFFECTS

Viscous, laminar mixed convection boundary-layer flow over a horizontal plate, with chemical reaction, is considered. The governing equations are expressed in nondimensional form. Group theory is employed to determine the invariant solutions of these equations under a particular continuous one-parameter group. Series solutions of the transformed coupled system of equations are then generated for velocity, temperature, and concentration functions using the Differential Transform Method (DTM) with Padé approximants. The influence of thermal buoyancy parameter, species buoyancy parameter, chemical reaction parameter, order of chemical reaction, Prandtl number, and Schmidt number on the flow characteristics is evaluated in detail The obtained solutions are verified by comparison with the numerical shooting quadrature results. Applications of the study arise in sheet materials processing, bio-reactors, and catalytic systems in chemical engineering.

The singular coupling limit for a simple pure number process

We consider the strongly continuous unitary one-parameter group on , which is related to the pure number quantum stochastic process restricted to the one-particle subspace . We approximate the Hamiltonian of the group by and go with g to . Then, the approximating unitary group converges to in the strong operator topology with a parameter given explicitly.

Operator ergodic theory for one-parameter decomposable groups

After initial treatment of the Fourier analysis and operator ergodic theory of strongly continuous decomposable one-parameter groups of operators in the Banach space setting, we show that in the setting of a super-reflexive Banach space X these groups automatically transfer from the setting of R to X the behavior of the Hilbert kernel, as well as the Fourier multiplier actions of functions of higher variation on R . These considerations furnish one-parameter groups with counterparts for the single operator theory in Berkson (2010) [4]. Since no uniform boundedness of one-parameter groups of operators is generally assumed in the present article, its results for the super-reflexive space setting go well beyond the theory of uniformly bounded one-parameter groups on UMD spaces (which was developed in Berkson et al., 1986 [13]), and in the process they expand the scope of vector-valued transference to encompass a genre of representations of R that are not uniformly bounded.

On strongly continuous one-parameter groups of automorphisms of multilinear functionals

We prove a structure theorem for strongly continuous one-parameter groups formed by surjective isometries of the space of bounded N-linear functionals over complex Hilbert spaces. As a consequence, the strongly continuous one-parameter automorphism groups of Cartan factors of type I are classified.

Bogoliubov Hamiltonians and one-parameter groups of Bogoliubov transformations

On the bosonic Fock space, a family of Bogoliubov automorphisms corresponding to a strongly continuous one-parameter group of symplectic maps (R(t))t∊R is considered. We give conditions that guarantee it to be implemented by a strongly continuous one-parameter group U(t) of unitary operators. The generator of such U(t) will be called a Bogoliubov Hamiltonian. Given (R(t))t∊R, a Bogoliubov Hamiltonian is defined up to an additive constant. We introduce two kinds of Bogoliubov Hamiltonians: type I, characterized by vanishing of the expectation value at the vacuum, and type II, characterized by the fact that its infimum equals zero. We give conditions so that they are well defined. We show that there exist cases when only HI is well defined, even though the classical Hamiltonian is positive (which may be interpreted as a kind of an infrared catastrophe), and when only HII is well defined (which means that one needs to introduce an infinite counterterm in the formula for the Hamiltonian).

Normalisers and centralisers of compact matrix groups. An elementary approach

Early investigations of operator stable laws and operator self-similar stochastic processes on a finite-dimensional vector space V = R d lead––under fullness assumption––to results of the following type: Given a continuous one-parameter group { exp ( t · E ) : t ∈ R } ⊆ GL ( V ) ( normalizing matrices) and a compact group K ⊆ GL ( V ) ( symmetries) such that exp( t · E) normalizes K for all t, then there exists a modification (exp( t · ( E + H)) = {exp( t · E) · exp( t · H)} centralizing K where {exp( t · H)} ⊆ K. ( E c := E + H is called commuting exponent.) It is challenging to obtain similar results in the context of operator semistable laws or operator semi-self-similar processes where the continuous one-parameter matrix group is replaced by a discrete group { a k : k ∈ Z } . Our aim is to provide elementary proofs of the following results—independently of the probabilistic background––which are interesting in their own right: (1) There exists a suitable power a k which is embeddable into a continuous one-parameter group { exp ( t · E ) : t ∈ R } belonging to the normaliser of K, and (2) there exists a shifted power b := a k · κ, κ ∈ K such that b is embeddable into a one-parameter group { exp ( t · E c ) : t ∈ R } belonging to the centraliser of K.

KMS states of quasi-free dynamics on Pimsner algebras

A continuous one-parameter group of unitary isometries of a right-Hilbert C ∗ -bimodule induces a quasi-free dynamics on the Cuntz–Pimsner C ∗ -algebra of the bimodule and on its Toeplitz extension. The restriction of such a dynamics to the algebra of coefficients of the bimodule is trivial, and the corresponding KMS states of the Toeplitz–Cuntz–Pimsner and Cuntz–Pimsner C ∗ -algebras are characterized in terms of traces on the algebra of coefficients. This generalizes and sheds light onto various earlier results about KMS states of the gauge actions on Cuntz algebras, Cuntz–Krieger algebras, and crossed products by endomorphisms. We also obtain a more general characterization, in terms of KMS weights, for the case in which the inducing isometries are not unitary, and accordingly, the restriction of the quasi-free dynamics to the algebra of coefficients is nontrivial.

Weighted Ergodic Theory And Dimension Free Estimates

Let T = {Ts}s∈R be a strongly continuous one-parameter group of positive operators acting on L(μ), where μ is a σ-finite measure and 1 ≤ p < ∞. We develop a weighted ergodic theory associated with T which parallels the classical theory of Ap-weights. A weighted transference principle is obtained in this setting and used to establish dimension free weighted inequalities for certain classical operators, including the higher order Riesz transforms.

Examples of One-Parameter Automorphism Groups¶of UHF Algebras

Examples of (strongly continuous) one-parameter groups of automorphisms of UHF C*-algebras are given. One example has the property that the infinitesimal generator does not have a union of finite-dimensional subalgebras as a core. (This solves the so-called core problem, raised by S. Sakai, in the negative.) The other has the property of not being classical or commutative, i.e., the one-parameter group is not just an inner perturbation of a one-parameter group which has an increasing sequence of invariant finite-dimensional subalgebras with dense union. (This was certainly expected but has not been proven explicitly.)

Semi-self-similar extremal processes

Let g be the distribution function (d.f.) of an extremal process Y. If g is invariant with respect to a continuous one-parameter group of time-space changes {ηα = (τα, Lα): α > 0}, i.e. g ∘ ηα = g ∀ α > 0, then g is self-similar. If g is invariant w.r.t. the cyclic group {η∘(n), n ∈Z} of a time-space change ν, then g is semi-self-similar. The semi-self-similar extremal processes are limiting for sequences of extremal processes Yn(t)=Ln−1 ∘ Y ∘ τn (t) if going along a geometrically increasing subsequence kn ∼ ϕn, ϕ > 1, n → ∞. The main properties of multivariate semi-self-similar extremal processes and some examples are discussed in the paper. The results presented are an analog of the theory of semi-self-similar processes with additive increments developed by Maejima and Sato in 1997.

Spectral Decompositions of Isometries oncp

The spatial action of the Stone-type spectral family for a certain type of strongly continuous one-parameter groups of surjective isometries, as well as of the spectral decomposition of their single elements oncp, 1≤p≤∞, is examined. The UMD structure, for 1<p<∞, and triangular truncations and duality forp=1, ∞ are involved. In various cases, concrete descriptions of the associated Riesz–Nagy operators and Arveson's spectral subspaces are derived as byproducts of independent interest.

On the spectral mapping theorem for one-parameter groups of operators

Let A be the generator of a strongly continuous nonquasianatytic oneparameter group of operators U(t) (A, in general, is unbounded). In this case one proves a spectral mapping theorem in the following form: . The proof is based on the theory of operators with separable spectrum.

Comparison of Commuting One-Parameter Groups of Isometries

Let $\alpha ,\;\beta$ be two commuting strongly continuous one-parameter groups of isometries on a Banach space $\mathcal {A}$ with generators ${\delta _\alpha }$ and ${\delta _\beta }$, and analytic elements $\mathcal {A}_\omega ^\alpha ,\;\mathcal {A}_\omega ^\beta$, respectively. Then it is easy to show that if ${\delta _\alpha }$ is relatively bounded by ${\delta _\beta }$, then $\mathcal {A}_\omega ^\beta \subseteq \mathcal {A}_\omega ^\alpha$, and in this paper we establish the inverse implication for unitary one-parameter groups on Hilbert spaces and for one-parameter groups of $^{\ast }$-automorphisms of abelian ${C^{\ast }}$-algebras. It is not known in general whether the inverse implication holds or not, but it does not hold for one-parameter semigroups of contractions.