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David Applebaum - One of the best experts on this subject based on the ideXlab platform.
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Operator-Valued Stochastic Differential Equations Arising from Unitary Group Representations
Journal of Theoretical Probability, 2001Co-Authors: David ApplebaumAbstract:Let π be a unitary representation of a Lie group G and (φ(t), t≥0) be a Lévy process in G. Using analytic vector techniques it is shown that the unitary process U(t)=π(φ(t)) satisfies an operator-valued stochastic differential equation. The prescription J(t) π(f)=U(t) π(f) U(t)* gives rise to an algebraic stochastic flow on the algebra generated by operators of the form π(f)=∫ f(g) π(g) dg where f is in the group algebra and dg is a Left Haar Measure. J(t) itself satisfies an operator-valued stochastic differential equation of a type which has been previously studied within the context of quantum stochastic calculus.
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Operator-Valued Stochastic Differential Equations Arising from Unitary Group Representations
Journal of Theoretical Probability, 2001Co-Authors: David ApplebaumAbstract:Let π be a unitary representation of a Lie group G and (φ(t), t≥0) be a Levy process in G. Using analytic vector techniques it is shown that the unitary process U(t)=π(φ(t)) satisfies an operator-valued stochastic differential equation. The prescription J(t) π(f)=U(t) π(f) U(t)* gives rise to an algebraic stochastic flow on the algebra generated by operators of the form π(f)=∫ f(g) π(g) dg where f is in the group algebra and dg is a Left Haar Measure. J(t) itself satisfies an operator-valued stochastic differential equation of a type which has been previously studied within the context of quantum stochastic calculus.
Saeid Maghsoudi - One of the best experts on this subject based on the ideXlab platform.
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A CONVOLUTION-INDUCED TOPOLOGY ON THE ORLICZ SPACE OF A LOCALLY COMPACT GROUP
Journal of The Australian Mathematical Society, 2015Co-Authors: Ibrahim Akbarbaglu, Saeid MaghsoudiAbstract:Let $G$ be a locally compact group with a fixed Left Haar Measure. In this paper, given a strictly positive Young function ${\rm\Phi}$ , we consider $L^{{\rm\Phi}}(G)$ as a Banach Left $L^{1}(G)$ -module. Then we equip $L^{{\rm\Phi}}(G)$ with the strict topology induced by $L^{1}(G)$ in the sense of Sentilles and Taylor. Some properties of this locally convex topology and a comparison with weak $^{\ast }$ , bounded weak $^{\ast }$ and norm topologies are presented.
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A strict topology on Orlicz spaces
Mathematische Nachrichten, 2014Co-Authors: Ibrahim Akbarbaglu, Saeid Maghsoudi, Juan B. Seoane-sepúlvedaAbstract:Let ϕ be a Young function, Ω be a locally compact space, and μ be a positive Radon Measure on Ω. We consider a strict topology βϕ (in the sense of Sentilles-Taylor) on the Orlicz function space Mϕ(Ω) and investigate various properties of this locally convex topology. We also study the Orlicz space Mϕ(G) of a locally compact group G with a Left Haar Measure under the strict topology βϕ and certain other natural locally convex topologies. Finally we present some results on various continuity properties of convolution operators on Mϕ(G) under the βϕ topology and other natural ones
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HYPERGROUP ALGEBRAS AS TOPOLOGICAL ALGEBRAS
Bulletin of the Australian Mathematical Society, 2014Co-Authors: Saeid Maghsoudi, Juan B. Seoane-sepúlvedaAbstract:Let K be a locally compact hypergroup endowed with a Left Haar Measure and let L1(K) be the usual Lebesgue space of K with respect to the Left Haar Measure. We investigate some properties of L1(K) under a locally convex topology β1. Among other things, the semireflexivity of (L1(K),β1) and of sequentiallyβ1-continuous functionals is studied. We also show that (L1(K),β1) with the convolution multiplication is always a complete semitopological algebra, whereas it is a topological algebra if and only if K is compact.
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Banach-Orlicz Algebras on a Locally Compact Group
Mediterranean Journal of Mathematics, 2013Co-Authors: Ibrahim Akbarbaglu, Saeid MaghsoudiAbstract:Let G be a locally compact group with a fixed Left Haar Measure λ. Given an N-function φ, we consider the Orlicz space \({L^{\varphi}(G)}\) under the convolution multiplication and establish that, for amenable groups under mild conditions on φ, it is a convolution algebra if and only if G is compact. Also we prove that for a locally compact group G, the convolution algebra \({L^{\varphi}(G)}\) has a bounded approximate identity if and only if G is discrete.
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On the Generalized Weighted Lebesgue Spaces of Locally Compact Groups
Abstract and Applied Analysis, 2011Co-Authors: Ibrahim Akbarbaglu, Saeid MaghsoudiAbstract:Let
Derek Dw Robinson - One of the best experts on this subject based on the ideXlab platform.
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Second-Order Subelliptic Operators on Lie Groups II: Real Measurable Principal Coefficients
Semigroups of Operators: Theory and Applications, 2000Co-Authors: Ter Afm Tom Elst, Derek Dw RobinsonAbstract:Let G be a connected Lie group with Lie algebra \(\mathfrak{g}\) and a 1,…,a d’ an algebraic basis of \(\mathfrak{g}\). Further let A i denote the generators of Left translations,acting on the L p -spaces L p (G; dg)formed with Left Haar Measure dg,in the directions a i . We consider second-order operators $$H = - \sum\limits_{{i,j = 1}}^{{d\prime }} {{{A}_{i}}{{c}_{{ij}}}{{A}_{j}} + \sum\limits_{{i = 1}}^{{d\prime }} {({{c}_{i}}{{A}_{i}} + {{A}_{i}}c_{i}^{\prime }) + {{c}_{0}}I} }$$ corresponding to a quadratic form with real measurable coeffcients c ij and complex c i , c ′ i , c 0 є L ∞. The matrix C = (c ij ) of principal coefficients, which is not necessarily symmetric, is assumed to satisfy the subellipticity condition $$\Re C = {{2}^{{ - 1}}}\Left( {C + {{C}^{*}}} \right) \geqslant \mu I > 0$$ uniformly over G.
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Second-order subelliptic operators on Lie groups III: Hölder continuous coefficients
Calculus of Variations and Partial Differential Equations, 1999Co-Authors: Ter Afm Tom Elst, Derek Dw RobinsonAbstract:Let G be a connected Lie group with Lie algebra \(\mathfrak{g}\) and \(a_1,\ldots,a_{d'}\) an algebraic basis of \(\mathfrak{g}\). Further let \(A_i\) denote the generators of Left translations, acting on the \(L_p\)-spaces \(L_p(G\,;dg)\) formed with Left Haar Measure dg, in the directions \(a_i\). We consider second-order operators \(\) corresponding to a quadratic form with complex coefficients \(c_{ij}\), \(c_{i}\), \(c'_{i}\), \(c_{0}\in L_{\infty}\). The principal coefficients \(c_{ij}\) are assumed to be Holder continuous and the matrix \(C=(c_{ij})\) is assumed to satisfy the (sub)ellipticity condition \(\) uniformly over G. We discuss the hierarchy relating smoothness properties of the coefficients of H with smoothness of the kernel. Moreover, we establish Gaussian type bounds for the kernel and its derivatives. Similar theorems are proved for operators \(\) in nondivergence form for which the principal coefficients are at least once differentiable.
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Second-order strongly elliptic operators on Lie groups with Hölder continuous coefficients
Journal of The Australian Mathematical Society, 1997Co-Authors: Ter Afm Tom Elst, Derek Dw RobinsonAbstract:Let G be a connected Lie group with Lie algebra g and a 1 , …, a d an algebraic basis of g. Further let A i denote the generators of Left translations, acting on the L p -spaces L p (G; dg) formed with Left Haar Measure dg , in the directions a i . We consider second-order operators in divergence form corresponding to a quadratic form with complex coefficients, bounded Holder continuous principal coefficients c ij and lower order coefficients c i , c′ i i , c 0 ∈ L ∞ such that the matrix C= (c ij ) of principal coefficients satisfies the subellipticity condition uniformly over G . We discuss the hierarchy relating smoothness properties of the coefficients of H with smoothness of the kernel and smoothness of the domain of powers of H on the L ρ -spaces. Moreover, we present Gaussian type bounds for the kernel and its derivatives. Similar theorems are proved for strongly elliptic operators in non-divergence form for which the principal coefficients are at least once differentiable.
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On positive Rockland operators
Colloquium Mathematicum, 1994Co-Authors: Pascal Auscher, Ter Afm Tom Elst, Derek Dw RobinsonAbstract:Let G be a homogeneous Lie group with a Left Haar Measure dg and L the action of G as Left translations on Lp(G; dg). Further, let H = dL(C) denote a homogeneous operator associated with L. If H is positive and hypoelliptic on L2 we prove that it is closed on each of the Lp-spaces, p e (1, 8), and that it generates a semigroup S with a smooth kernel K which, with its derivatives, satisfies Gaussian bounds. The semigroup is holomorphic in the open right half-plane on all the Lp-spaces, p e [1, 8]. Further extensions of these results to nonhomogeneous operators and general representations are also given.
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Lp-regularity of subelliptic operators on Lie groups
Journal of Operator Theory, 1992Co-Authors: Rj Robert Burns, A. F. M. Ter Elst, Derek Dw RobinsonAbstract:Let G be a Lie group with a Left Haar Measure dg and let L denote the action of G as Left translations on Lp ( G; dg). If at, ... ,ad' are elements of the Lie algebra g of G and Ai = dL(ai) the generators of the corresponding one-parameter subgroups t 1-+ L(exp(tai)) define the en-subspace L~;n as the common domain of all n-th order monomials Mn in the A j and introduce the norm II . 1I~;n on L~;n by where the supremum is over all monomials of order k ~ n. Then define d' d' H = L cjjAjAj +L cjA j i,j=t i=t with domain D(H) = L~;2' where Cij, Cj E C and the real part of the matrix C = (Cij) is strictly positive-definite. We establish that for each P E (1,00), n E N and all large positive A the spaces L~;n and D((AI +H)n/2) coincide and there is a Cp,n,>. > 0 such that for all
Ibrahim Akbarbaglu - One of the best experts on this subject based on the ideXlab platform.
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A CONVOLUTION-INDUCED TOPOLOGY ON THE ORLICZ SPACE OF A LOCALLY COMPACT GROUP
Journal of The Australian Mathematical Society, 2015Co-Authors: Ibrahim Akbarbaglu, Saeid MaghsoudiAbstract:Let $G$ be a locally compact group with a fixed Left Haar Measure. In this paper, given a strictly positive Young function ${\rm\Phi}$ , we consider $L^{{\rm\Phi}}(G)$ as a Banach Left $L^{1}(G)$ -module. Then we equip $L^{{\rm\Phi}}(G)$ with the strict topology induced by $L^{1}(G)$ in the sense of Sentilles and Taylor. Some properties of this locally convex topology and a comparison with weak $^{\ast }$ , bounded weak $^{\ast }$ and norm topologies are presented.
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A strict topology on Orlicz spaces
Mathematische Nachrichten, 2014Co-Authors: Ibrahim Akbarbaglu, Saeid Maghsoudi, Juan B. Seoane-sepúlvedaAbstract:Let ϕ be a Young function, Ω be a locally compact space, and μ be a positive Radon Measure on Ω. We consider a strict topology βϕ (in the sense of Sentilles-Taylor) on the Orlicz function space Mϕ(Ω) and investigate various properties of this locally convex topology. We also study the Orlicz space Mϕ(G) of a locally compact group G with a Left Haar Measure under the strict topology βϕ and certain other natural locally convex topologies. Finally we present some results on various continuity properties of convolution operators on Mϕ(G) under the βϕ topology and other natural ones
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Banach-Orlicz Algebras on a Locally Compact Group
Mediterranean Journal of Mathematics, 2013Co-Authors: Ibrahim Akbarbaglu, Saeid MaghsoudiAbstract:Let G be a locally compact group with a fixed Left Haar Measure λ. Given an N-function φ, we consider the Orlicz space \({L^{\varphi}(G)}\) under the convolution multiplication and establish that, for amenable groups under mild conditions on φ, it is a convolution algebra if and only if G is compact. Also we prove that for a locally compact group G, the convolution algebra \({L^{\varphi}(G)}\) has a bounded approximate identity if and only if G is discrete.
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On the Generalized Weighted Lebesgue Spaces of Locally Compact Groups
Abstract and Applied Analysis, 2011Co-Authors: Ibrahim Akbarbaglu, Saeid MaghsoudiAbstract:Let
Benjamin Willson - One of the best experts on this subject based on the ideXlab platform.
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A fixed point theorem and the existence of a Haar Measure for hypergroups satisfying conditions related to amenability
Canadian Mathematical Bulletin, 2015Co-Authors: Benjamin WillsonAbstract:In this paper we present a fixed point property for amenable hypergroups which is analogous to Rickert's fixed point theorem for semigroups. It equates the existence of a Left invariant mean on the space of weakly right uniformly continuous functions to the existence of a fixed point for any action of the hypergroup. Using this fixed point property, a certain class of hypergroups are shown to have a Left Haar Measure.