We consider positive semidefinite kernels valued in the $*$-algebra of adjointable operators on a VE-space (Vector Euclidean space) and that are invariant under actions of $*$-semigroups. A rather general dilation theorem is stated and proved: for these kind of kernels, representations of the $*$-semigroup on either the VE-spaces of linearisation of the kernels or on their reproducing kernel VE-spaces are obtainable. We point out the reproducing kernel fabric of dilation theory and we show that the general theorem unifies many dilation results at the non topological level.
We show that pure strongly continuous semigroups of adjointable isometries on a Hilbert C*-module are standard right shifts. By counter examples, we illustrate that the analogy of this result with the classical result on Hilbert spaces by Sz.-Nagy, cannot be improved further to understand arbitrary isometry semigroups of isometries in the classical way.
The geometry of spaces with indefinite inner product, known also as Krein spaces, is a basic tool for developing Operator Theory therein. In the present paper we establish a link between this geometry and the algebraic theory of *-semigroups. It goes via the positive definite functions and related to them reproducing kernel Hilbert spaces. Our concern is in describing properties of elements of the semigroup which determine shift operators which serve as Pontryagin fundamental symmetries
Motivated by applications to transition semigroups, we introduce the notion of a norming dual pair and study a Pettis-type integral on such pairs. In particular, we establish a sufficient condition for integrability. We also introduce and study a class of semigroups on such dual pairs which are an abstract version of transition semigroups. Using our results, we give conditions ensuring that a semigroup consisting of kernel operators has a Laplace transform which also consists of kernel operators. We also provide conditions under which a semigroup is uniquely determined by its Laplace transform…
The aim of this paper is to present a unified framework in the setting of Hilbert $C^*$-modules for the scalar- and vector-valued reproducing kernel Hilbert spaces and $C^*$-valued reproducing kernel spaces. We investigate conditionally negative definite kernels with values in the $C^*$-algebra of adjointable operators acting on a Hilbert $C^*$-module. In addition, we show that there exists a two-sided connection between positive definite kernels and reproducing kernel Hilbert $C^*$-modules. Furthermore, we explore some conditions under which a function is in the reproducing kernel module and …
We prove a perturbation result for positive semigroups, thereby extending a heat kernel estimate by Barlow, Grigor'yan and Kumagai for Dirichlet forms (\cite{bgk2009}) to positive semigroups. This also leads to a generalization of domination for semigroups on $L_p$-spaces.
M. Embry and A. Lambert initiated the study of a semigroup of operators $\{S_t\}$ indexed by a non-negative real number $t$ and termed it as weighted translation semigroup. The operators $S_t$ are defined on $L^2(\mathbb R_+)$ by using a weight function. The operator $S_t$ can be thought of as a continuous analogue of a weighted shift operator. In this paper, we show that every left invertible operator $S_t$ can be modeled as a multiplication by $z$ on a reproducing kernel Hilbert space $\cal H$ of vector-valued analytic functions on a certain disc centered at the origin and the reproducing ke…
It is well known that Nash-type inequalities for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper bounds for the associated symmetric Markov semigroups. In this paper, we show that both imply (and hence are equivalent to) off-diagonal heat kernel upper bounds under some mild assumptions. Our approach is based on a new generalized Davies' method. Our results extend that of \cite{CKS} for Nash-type inequalities with power order considerably and also extend that of \cite{Gri} for second order differential operators on a complete non-compact manifold.
We give a characterization of a variation of constants type estimate relating two positive semigroups on (possibly different) $L_p$-spaces to one another in terms of corresponding estimates for the respective generators and of estimates for the respective resolvents. The results have applications to kernel estimates for semigroups induced by accretive and non-local forms on $σ$-finite measure spaces.
On $\mathbb R^N$ equipped with a normalized root system $R$, a multiplicity function $k(α) > 0$, and the associated measure $$ dw(\mathbf x)=\prod_{α\in R}|\langle \mathbf x,α\rangle|^{k(α)}\, d\mathbf x, $$ we consider a Dunkl Schrödinger operator $L=-Δ_k+V$, where $Δ_k$ is the Dunkl Laplace operator and $V\in L^1_{\rm loc} (dw)$ is a non-negative potential. Let $h_t(\mathbf x,\mathbf y)$ and $k^{\{V\}}_t(\mathbf x,\mathbf y)$ denote the Dunkl heat kernel and the integral kernel of the semigroup generated by $-L$ respectively. We prove that $k^{\{V\}}_t(\mathbf x,\mathbf y)$ satisfies the fol…
We prove that every bounded, positive, irreducible, stochastically continuous semigroup on the space of bounded, measurable functions which is strong Feller, consists of kernel operators and possesses an invariant measure converges pointwise. This differs from Doob's theorem in that we do not require the semigroup to be Markovian and request a fairly weak kind of irreducibility. In addition, we elaborate on the various notions of kernel operators in this context, show the stronger result that the adjoint semigroup converges strongly and discuss as an example diffusion equations on rough domain…
The Weil representation of a real symplectic group $Sp(2n,R)$ admits a canonical extension to a holomorphic representation of a certain complex semigroup consisting of Lagrangian linear relations (this semigroup includes the Olshanski semigroup). We obtain the explicit realization of the Weil representation of this semigroup in the Cartier model, i.e., in the space of smooth sections of a certain line bundle on the $2n$-dimensional torus $T^{2n}$. We show that operators of the representation are integral operators whose kernels are theta-functions on $T^{4n}$.
Large time behaviour of heat semigroups (and more generally, of positive selfadjoint semigroups) is studied. Convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoint semigroups. This framework includes Laplacians on manifolds, metric graphs and discrete graphs.
We give a modified definition of a reproducing kernel Hilbert $C^*$-module (shortly, $RKHC^*M$) without using the condition of self-duality and discuss some related aspects; in particular, an interpolation theorem is presented. We investigate the exterior tensor product of $RKHC^*M$s and find their reproducing kernel. In addition, we deal with left multipliers of $RKHC^*M$s. Under some mild conditions, it is shown that one can make a new $RKHC^*M$ via a left multiplier. Moreover, we introduce the Berezin transform of an operator in the context of $RKHC^*M$s and construct a unital subalgebra of…
By suitably extending a Feynman-Kac formula of Simon [Canadian Math. Soc. Conf. Proc, 28 (2000), 317-321], we study one-parameter semigroups generated by (the negative of) rather general Schroedinger operators, which may be unbounded from below and include a magnetic vector potential. In particular, a common domain of essential self-adjointness for such a semigroup is specified. Moreover, each member of the semigroup is proven to be a maximal Carleman operator with a continuous integral kernel given by a Brownian-bridge expectation. The results are used to show that the spectral projections of…
Nash and Sobolev inequalities are known to be equivalent to ultracontractive properties of heat-like Markov semigroups, hence to uniform on-diagonal bounds on their kernel densities. In non ultracontractive settings, such bounds can not hold, and (necessarily weaker, non uniform) bounds on the semigroups can be derived by means of weighted Nash (or super-Poincaré) inequalities. The purpose of this note is to show how to check these weighted Nash inequalities in concrete examples, in a very simple and general manner. We also deduce off-diagonal bounds for the Markov kernels of the semigroups, r…
Given a positive, irreducible and bounded C_0-semigroup on a Banach lattice with order continuous norm, we prove that the peripheral point spectrum of its generator is trivial whenever one of its operators dominates a non-trivial compact or kernel operator. For a discrete semigroup, i.e. for powers of a single operator T, we show that the point spectrum of some power T^k intersects the unit circle at most in 1. As a consequence, we obtain a sufficient condition for strong convergence of the C_0-semigroup and for a subsequence of the powers of T, respectively.
We introduce a class of central symmetric infinitely divisible probability measures on compact Lie groups by lifting the characteristic exponent from the real line via the Casimir operator. The class includes Gauss, Laplace and stable-type measures. We find conditions for such a measure to have a smooth density and give examples. The Hunt semigroup and generator of convolution semigroups of measures are represented as pseudo-differential operators. For sufficiently regular convolution semigroups, the transition kernel has a tractable Fourier expansion and the density at the neutral element may…
In this paper we characterize the discrete Hölder spaces by means of the heat and Poisson semigroups associated to the discrete Laplacian. These characterizations allow us to get regularity properties of fractional powers of the discrete Laplacian and the Bessel potentials along these spaces and also in the discrete Zygmund spaces in a more direct way than using the pointwise definition of the spaces. To obtain our results, it has been crucial to get boundedness properties of the heat and Poisson kernels and their derivatives in both space and time variables. We believe that these estimates ar…
Convoluted $C$-cosine functions and semigroups in a Banach space setting extending the classes of fractionally integrated $C$-cosine functions and semigroups are systematically analyzed. Structural properties of such operator families are obtained. Relations between convoluted $C$-cosine functions and analytic convoluted $C$-semigroups, introduced and investigated in this paper are given through the convoluted version of the abstract Weierstrass formula which is also proved in the paper. Ultradistribution and hyperfunction sines are connected with analytic convoluted semigroups and ultradistri…
In this paper we study hermitian kernels invariant under the action of a semigroup with involution. We characterize those hermitian kernels which realize the given action by bounded operators on a Krein space. Applications to the GNS representation of *-algebras associated to hermitian functionals are given. We explain the key role played by the Kolmogorov decomposition in the construction of Weyl exponentials associated to an indefinite inner product and in the dilation thoery of hermitian maps on C*-algebras.
We present new conditions for semigroups of positive operators to converge strongly as time tends to infinity. Our proofs are based on a novel approach combining the well-known splitting theorem by Jacobs, de Leeuw and Glicksberg with a purely algebraic result about positive group representations. Thus we obtain convergence theorems not only for one-parameter semigroups but for a much larger class of semigroup representations. Our results allow for a unified treatment of various theorems from the literature that, under technical assumptions, a bounded positive $C_0$-semigroup containing or dom…
Let $\{T(t)\}_{t\ge 0}$ be a $C_0$-semigroup on a separable Hilbert space $H$. We characterize that $T(t)$ is an $m$-isometry for every $t$ in terms that the mapping $t\in \Bbb R^+ \rightarrow \|T(t)x\|^2$ is a polynomial of degree less than $m$ for each $x\in H$. This fact is used to study $m$-isometric right translation semigroup on weighted $L^p$-spaces. We characterize the above property in terms of conditions on the infinitesimal generator operator or in terms of the cogenerator operator of $\{ T(t)\}_{t\geq 0}$. Moreover, we prove that a non-unitary $2$-isometry on a Hilbert space satisf…
We consider Feller semigroups of operators determinated by systems of jumps dominated by the rotation invariant stable Lévy measure. Using an approximation schema we prove the existence and obtain estimates of corresponding heat kernels.
Completely simple semigroups arise as the support of limiting measures of random walks on semigroups. Such a limiting measure is supported on the kernel of the semigroup. Forming tensor powers of the random walk leads to a hierarchy of the limiting kernels. Tensor squares lead to quantum observables on the kernel. Recall that zeons are bosons modulo the basis elements squaring to zero. Using zeon powers leads naturally to quantum observables which reveal the structure of the kernel. Thus asymptotic information about the random walk is related to algebraic properties of the zeon powers of the r…
We study small time bounds for transition densities of convolution semigroups corresponding to pure jump Lévy processes in $\mathbb{R}^{d}$, $d \geq 1$, including those with jumping kernels exponentially and subexponentially localized at infinity. For a large class of Lévy measures, non-necessarily symmetric nor absolutely continuous with respect to the Lebesgue measure, we find the optimal, both in time and space, upper bound for the corresponding transition kernels at infinity. In case of Lévy measures that are symmetric and absolutely continuous, with densities $g$ such that $g(x) \asymp f(…
In this paper we prove the global in time solvability of the continuous growth--fragmentation--coagulation equation with unbounded coagulation kernels, in spaces of functions having finite moments of sufficiently high order. The main tool is the recently established result on moment regularization of the linear growth--fragmentation semigroup that allows us to consider coagulation kernels whose growth for large clusters is controlled by how good the regularization is, in a similar manner to the case when the semigroup is analytic.
We present a general approach to proving the existence of spectral gaps and asynchronous exponential growth for growth-fragmentation semigroups in moment spaces $L^{1}(\mathbb{R}_{+};\ x^{α}dx)$ and $L^{1}(\mathbb{R} _{+};\ \left( 1+x\right) ^{α}dx)$ for unbounded total fragmentation rates and continuous growth rates $r(.)$\ such that $\int_{0}^{+\infty } \frac{1}{r(τ)}dτ=+\infty .\ $The analysis is based on weak compactness tools and Frobenius theory of positive operators and holds provided that $α>\widehat{α}$ for a suitable threshold $\widehat{ α}\geq 1$ that depends on the moment space we …
We present a simple observation showing that the heat kernel on a locally finite graph behaves for short times $t$ roughly like $t^d$, where $d$ is the combinatorial distance. This is very different from the classical Varadhan type behavior on manifolds. Moreover, this also gives that short time behavior and global behavior of the heat kernel are governed by two different metrics whenever the degree of the graph is not uniformly bounded.
We systematize and generalize recent results of Gerlach and Glück on the strong convergence and spectral theory of bounded (positive) operator semigroups $(T_s)_{s\in S}$ on Banach spaces (lattices). (Here, $S$ can be an arbitrary commutative semigroup, and no topological assumptions neither on $S$ nor on its representation are required.) To this aim, we introduce the "semigroup at infinity" and give useful criteria ensuring that the well-known Jacobs--de Leeuw--Glicksberg splitting theory can be applied to it. Next, we confine these abstract results to positive semigroups on Banach lattices w…
This paper considers universal Hilbert space operators in the sense of Rota, and gives criteria for universality of semigroups in the context of uniformly continuous semigroups and contraction semigroups. Specific examples are given. Universal semigroups provide models for these classes of semigroups: following a line of research initiated by Shimorin, models for concave semigroups are developed, in terms of shifts on reproducing kernel Hilbert spaces.
The purpose of this article is to establish upper and lower estimates for the integral kernel of the semigroup exp(-tP) associated to a classical, strongly elliptic pseudodifferential operator P of positive order on a closed manifold. The Poissonian bounds generalize those obtained for perturbations of fractional powers of the Laplacian. In the selfadjoint case, extensions to t in C_+ are studied. In particular, our results apply to the Dirichlet-to-Neumann semigroup.
We study a spatial asymptotic behaviour at infinity of kernels $p_t(x)$ for convolution semigroups of nonlocal pseudo-differential operators. We give general and sharp sufficient conditions under which the limits $$ \lim_{r \to \infty} \frac{p_t(rθ-y)}{t \, ν(rθ)}, \quad t \in T, \ \ θ\in E, \ \ y \in \mathbb R^d, $$ exist and can be effectively computed. Here $ν$ is the corresponding Lévy density, $T \subset (0,\infty)$ is a bounded time-set and $E$ is a subset of the unit sphere in $\mathbb R^d$, $d \geq 1$. Our results are local on the unit sphere. They apply to a wide class of convolution …
We prove a Weiss conjecture on $β$-admissibility of control and observation operators for discrete and continuous $γ$-hypercontractive semigroups of operators, by representing them in terms of shifts on weighted Bergman spaces and using a reproducing kernel thesis for Hankel operators. Particular attention is paid to the case $γ=2$, which corresponds to the unweighted Bergman shift.
We establish convolution inequalities for Besov spaces $B_{p,q}^s$ and Triebel--Lizorkin spaces $F_{p,q}^s$. As an application, we study the mapping properties of convolution semigroups, considered as operators on the function spaces $A_{p,q}^s$, $A \in \{B,F\}$. Our results apply to a wide class of convolution semigroups including the Gauß--Weierstraß semigroup, stable semigroups and heat kernels for higher-order powers of the Laplacian $(-Δ)^m$, and we can derive various caloric smoothing estimates.
We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Under the 2-Poincaré inequality and a weak Bakry-Émery curvature type condition, this BV class is identified with the heat semigroup based Besov class $\mathbf{B}^{1,1/2}(X)$ that was introduced in our previous paper. Assuming furthermore a quasi Bakry-Émery curvature type condition, we identify the Sobolev class $W^{1,p}(X)$ with $\mathbf{B}^{p,1/2}(X)$ for $p>1$. Consequences of those identifications in terms of isoperimetric and Sobolev inequalities wit…
We investigate $L^p(γ)$-$L^q(γ)$ off-diagonal estimates for the Ornstein-Uhlenbeck semigroup $(e^{tL})_{t > 0}$. For sufficiently large $t$ (quantified in terms of $p$ and $q$) these estimates hold in an unrestricted sense, while for sufficiently small $t$ they fail when restricted to maximal admissible balls and sufficiently small annuli. Our counterexample uses Mehler kernel estimates.
In this paper we investigate the class of invariant positive definite kernels on the free semigroup on N generators. We provide a combinatorial description of the positivity of the kernel in terms of Dyck paths and then we find a displacement equation that encodes the invariance property of the kernel.
With a view toward fractal spaces, by using a Korevaar-Schoen space approach, we introduce the class of bounded variation (BV) functions in a general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry-Émery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global $L^1$ Sobolev embeddings and isope…
For an increasing sequence $(T_n)$ of one-parameter semigroups of sub Markovian kernel operators over a Polish space, we study the limit semigroup and prove sufficient conditions for it to be strongly Feller. In particular, we show that the strong Feller property carries over from the approximating semigroups to the limit semigroup if the resolvent of the latter maps the constant 1 function to a continuous function. This is instrumental in the study of elliptic operators on $\mathbb{R}^d$ with unbounded coefficients: our abstract result enables us to assign a semigroup to such an operator and …
We give the upper and the lower estimates of heat kernels for Schrödinger operators $H=-Δ+V$, with nonnegative and locally bounded potentials $V$ in $\mathbb{R}^d$, $d \geq 1$. We observe a factorization: the contribution of the potential is described separately for each spatial variable, and the interplay between the spatial variables is seen only through the Gaussian kernel - optimal in the lower bound and nearly optimal in the upper bound. In some regimes we observe the exponential decay in time with the rate corresponding to the bottom of the spectrum of $H$. Our estimates identify in a fa…
We study heat kernels of Schrödinger operators whose kinetic terms are non-local operators built for sufficiently regular symmetric Lévy measures with radial decreasing profiles and potentials belong to Kato class. Our setting is fairly general and novel -- it allows us to treat both heavy- and light-tailed Lévy measures in a joint framework. We establish a certain relative-Kato bound for the corresponding semigroups and potentials. This enables us to apply a general perturbation technique to construct the heat kernels and give sharp estimates of them. Assuming that the Lévy measure and the po…
On a doubling metric measure space endowed with a "carré du champ", we consider $L^p$ estimates $(G_p)$ of the gradient of the heat semigroup and scale-invariant $L^p$ Poincaré inequalities $(P_p)$. We show that the combination of $(G_p)$ and $(P_p)$ for $p\ge 2$ always implies two-sided Gaussian heat kernel bounds. The case $p=2$ is a famous theorem of Saloff-Coste, of which we give a shorter proof, without parabolic Moser iteration. We also give a more direct proof of the main result in \cite{HS}. This relies in particular on a new notion of $L^p$ Hölder regularity for a semigroup and on a c…
Consider the Schrödinger operator $ \mathcal L^V=-Δ+V $ on $\R^d$, where $V:\R^d\to [0,\infty)$ is a nonnegative and locally bounded potential on $\R^d$ so that for all $x\in \R^d$ with $|x|\ge 1$, $c_1g(|x|)\le V(x)\le c_2g(|x|)$ with some constants $c_1,c_2>0$ and a nondecreasing and strictly positive function $g:[0,\infty)\to [1,+\infty)$ that satisfies $g(2r)\le c_0 g(r)$ for all $r>0$ and $\lim_{r\to \infty} g(r)=\infty.$ We establish global in time and qualitatively sharp bounds for the heat kernel of the associated Schrödinger semigroup by the probabilistic method. In particular, we can…
On a doubling metric measure space $(M,d,μ)$ endowed with a "carré du champ", let $\mathcal{L}$ be the associated Markov generator and $\dot L^{p}_α(M,\mathcal{L},μ)$ the corresponding homogeneous Sobolev space of order $0<α<1$ in $L^p$, $1<p<+\infty$, with norm $\left\|\mathcal{L}^{α/2}f\right\|_p$. We give sufficient conditions on the heat semigroup $(e^{-t\mathcal{L}})_{t>0}$ for the spaces $\dot L^{p}_α(M,\mathcal{L},μ) \cap L^\infty(M,μ)$ to be algebras for the pointwise product. Two approaches are developed, one using paraproducts (relying on extrapolation to prove their boundedness) and…
Let $Ω\subset {\bf R}^d$ be open. We investigate conditions under which an operator $T$ on $L_2(Ω)$ has a continuous kernel $K \in C(\overline Ω\times \overline Ω)$. In the centre of our interest is the condition $T L_2(Ω) \subset C(\overline Ω)$, which one knows for many semigroups generated by elliptic operators. This condition implies that $T^3$ has a kernel in $C(\overline Ω\times \overline Ω)$ if $T$ is self-adjoint and $Ω$ is bounded, and the power $3$ is best possible. We also analyse Mercer's theorem in our context.
Let (M,μ) be a sigma-finite measure space. Let (T_t) be a semigroup of positive preserving maps on (M,μ) with standard assumptions. We prove a H_1-BMO duality theory with assumptions only on T_t. The BMO is defined as spaces of functions f such that the L_\infty norm of sup_tT_t|f-T_tf|^2 is finite. The H1 is defined by square functions of P. A. Meyer's gradient form. Our argument does not rely on any geometric/metric structure of M nor on the kernel of the semigroups of operators. This abstract argument allows to extend our main results to the noncommutative setting, e.g. the case where L_\in…
