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Abstract: Abstract Let \({\mathbb {H}}_d\) ( \(d\in {\mathbb {N}}^*\) ) be the \((2d+1)\) -dimensional Heisenberg group and let K be a compact connected subgroup of \(\text{Aut}({\mathbb {H}}_d)\) acting in the usual way on \({\mathbb {H}}_d.\) In this work, we define the semidirect product \(G\,{:}{=}\,K\ltimes {\mathbb {H}}_d,\) for which \((K,{\mathbb {H}}_d)\) is a Gelfand pair. It is well-known in the representation theory of Lie groups (theory of orbit method) that the unitary dual \({\widehat{G}}\) of G is in one to one correspondence with the space of admissible coadjoint orbits \({\mathfrak {g}}^\ddagger /G\) (see, [19]). The aim of this paper, is to give a nice description of the topology of \({\mathfrak {g}}^\ddagger /G\) and we show that the dual topology of G could be read from the quotient topology of \({\mathfrak {g}}^\ddagger /G.\) More precisely, we prove that the Kirillov–Lipsman’s (orbit mapping) bijection $$\begin{aligned} {\widehat{G}} \simeq {\mathfrak {g}}^\ddagger /G \end{aligned}$$ is a homeomorphism. This is a generalization of the previous result obtained in [12]. PubDate: 2022-11-15

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Abstract: Abstract In this paper, we give criteria for the relatively compact sets of Banach space-valued bounded-variation spaces in the sense of Jordan and Banach space-valued bounded Wiener p-variation spaces as \(p\in (0,1)\) . Then, we give sufficient conditions for the relatively compact sets of others Banach space-valued bounded variation spaces, such as Banach space-valued bounded Wiener p-variation spaces as \(p\in (1,\infty )\) , Banach space-valued bounded Wiener–Young variation spaces, Banach space-valued bounded Schramm variation spaces, Banach space-valued bounded Waterman variation spaces, Banach space-valued bounded Riesz variation spaces, and Banach space-valued bounded Korenblum variation spaces. PubDate: 2022-11-14

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Abstract: Abstract A Saphar operator T on a Banach space X is one whose kernel is contained in its generalized range \(\bigcap \nolimits _{i=1}^{\infty }{\mathcal {R}}(T^{n})\) and its range and kernel are closed complemented subspaces. The purpose of this paper is to introduce and study a new class of operators that subsumes the class of Saphar operators, namely, the class of generalized Saphar operators. It is shown that the operators introduced can be characterized in several ways, particularly by means of specific Kato decompositions. Generalized Saphar spectrum of an operator T is also introduced and proven to be a compact subset of the complex plane. PubDate: 2022-11-08

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Abstract: Abstract For a nonempty compact subset \(\sigma\) in the plane, the space \(AC(\sigma )\) is the closure of the space of complex polynomials in two real variables under a particular variation norm. In the classical setting, AC[0, 1] contains several other useful dense subsets, such as continuous piecewise linear functions, \(C^1\) functions and Lipschitz functions. In this paper, we examine analogues of these results in this more general setting. PubDate: 2022-11-07

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Abstract: Abstract We consider the well-posedness and blowup criterion to the double-diffusive magnetoconvection system in 3D. First, we establish the existence and uniqueness of the local strong solution to the system (1.1) in \(H^1({\mathbb {R}}^3)\) with arbitrary initial data, and obtain the global strong solution when the \(L^2\) norm of the initial data is small. This result can be regarded as a generalization of the methods in Chen et al. (J Math Phys 60(1):011511, 2019). Then, we provide some sufficient conditions for the breakdown of local strong solution to system (1.1) in terms of velocity (or gradient of velocity) in weak \(L^p\) spaces. Finally, we focus on blowup criterion only depends on partial derivative of the planar components \(({\tilde{u}}, {\tilde{b}})\) without \(u_3\) and \(b_3\) in \({{\text {BMO}}}^{-1}\) space. More precisely, if local solution satisfies $$\begin{aligned} \int ^{T}_{0}\Vert \nabla _{h}{\tilde{u}}(t)\Vert ^{2}_{{\text {BMO}}^{-1}} +\Vert \nabla _{h}{\tilde{b}}(t)\Vert ^{2}_{{\text {BMO}}^{-1}} {\text {d}}t<\infty , \end{aligned}$$ then the strong solution \((u, b, \theta , s)\) can be extended smoothly beyond \(t=T\) . This improves and extends several previous BKM’s criteria (Chol-Jun in Nonlinear Anal Real World Appl 59:103271, 2021; Guo et al. in J Math Anal Appl 458(1):755–766, 2018). PubDate: 2022-11-03

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Abstract: Abstract In this paper, we establish necessary and sufficient conditions under which certain weighted inequalities hold for a class of quasilinear integral operators with kernels. PubDate: 2022-10-29

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Abstract: Abstract We initiate a study of ternary rings of unbounded operators (TRUOs) which are local version of ternary rings of bounded operators. Abstract definition of ternary Pro \(C^*\) -rings is also proposed. A one-to-one correspondence between representations of TRUO and its linking Pro \(C^*\) -algebra is obtained. Finally we examine their tensor products particularly, injective and projective tensor products. PubDate: 2022-10-26

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Abstract: Abstract The goal of this paper is to investigate three kinds of stability of pullback random attractors (PRAs) for stochastic nonclassical diffusion equations with distributed delay and constant delay perturbed by operator-type noise defined on \({\mathbb{R}}^{n}.\) We first prove the existence, uniqueness, backward compactness and backward longtime stability of PRAs for this equation. We then establish the zero-memory stability of PRAs. Finally, we study the asymptotically autonomous stability of PRAs. Due to the problem of the non-compactness of Sobolev embeddings on \({\mathbb{R}}^{n},\) we use the backward tail-estimates method and spectrum decomposition technique to prove the backward asymptotic compactness of solutions. PubDate: 2022-10-15

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Abstract: Abstract In this article, the authors investigate the differentiability of functions from logarithmic Besov spaces via establishing the high order analogue of the Lebesgue differentiation theorem for these functions, and prove that the related exceptional sets have zero logarithmic Besov capacity. To this end, the authors establish the lifting theorem and the Peetre maximal function characterization of logarithmic Besov spaces, and then give several properties for the logarithmic Besov capacity, including delicately estimating the logarithmic Besov capacity of Euclidean balls, which might have their own independent interests. PubDate: 2022-10-11

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Abstract: Abstract We define positive Toeplitz operators between harmonic Bergman–Besov spaces \(b^p_\alpha \) on the unit ball of \({\mathbb {R}}^n\) for the full ranges of parameters \(0<p<\infty \) , \(\alpha \in {\mathbb {R}}\) . We give characterizations of bounded and compact Toeplitz operators taking one harmonic Bergman–Besov space into another in terms of Carleson and vanishing Carleson measures. We also give characterizations for a positive Toeplitz operator on \(b^{2}_{\alpha }\) to be a Schatten class operator \(S_{p}\) in terms of averaging functions and Berezin transforms for \(1\le p<\infty \) , \(\alpha \in {\mathbb {R}}\) . Our results extend those known for harmonic weighted Bergman spaces. PubDate: 2022-10-05

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Abstract: Abstract It is known that every compact linear operator from a Banach space to a uniform algebra can be approximated by norm attaining ones. In this paper, we study this approximation for compact operators whose range is a vector-valued function space. PubDate: 2022-10-02

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Abstract: Abstract We introduce the notion centre of a convex set and study the space of continuous affine functions on a compact convex set with a centre. We show that these spaces are precisely the dual of a base normed space in which the underlying base has a (unique) centre. We also characterize the corresponding base norm space. We obtain a condition on a compact, balanced, convex subset of a locally convex space, so that the corresponding space of continuous affine functions on the convex set is an absolute order unit space. Similarly, we characterize a condition on the base with a centre of a base normed space, so that the latter becomes an absolutely base normed space. PubDate: 2022-10-01

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Abstract: Abstract We study the class of compact spaces that appear as structure spaces of separable Banach lattices. In other words, we analyze what C(K) spaces appear as principal ideals of separable Banach lattices. Among other things, it is shown that every such compactum K admits a strictly positive regular Borel measure of countable type that is analytic, and in the nonmetrizable case these compacta are saturated with copies of \(\beta {{\mathbb{N}}}.\) Some natural questions about this class are left open. PubDate: 2022-09-30

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Abstract: Abstract We give the necessary and sufficient conditions for bounded multilinear operators on \(X_{1}\times \cdots \times X_{k}\times c_{0}\) to be nuclear or multiple 1-summing. We find the necessary and sufficient conditions for some bounded bilinear operators from \(l_{p}\times c_{0}\) into Y to be nuclear or multiple 1-summing operators. In the case of some bounded bilinear operators from \(l_{p}\times c_{0}\) into \(c_{0}\) associated to the classical methods of summability, we find the necessary and sufficient conditions for these be nuclear or multiple 1-summing operators. We show that, contrary to the bilinear case, in the multilinear case the situation change. PubDate: 2022-09-25

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Abstract: Abstract Let \({\mathbb{M}}=P\times {M}\) be a variable Mautner group. We describe the \(C^*\) -algebra \(C^*({\mathbb{M}})\) of \({\mathbb{M}}\) in terms of an algebra of operator fields defined over \(P\times {{\mathbb{C}}^2} .\) PubDate: 2022-09-24

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Abstract: Abstract In this paper, we introduce actions of fusion algebras on unital \(C^*\) -algebras, and define amenability for fusion algebraic actions. Motivated by S. Neshveyev et al.’s work, considering the representation ring of a compact quantum group as a fusion algebra, we define the canonical fusion algebraic (for short, CFA) form of a discrete quantum group action on a compact quantum space. Furthermore, through the CFA form, we define FA-amenability of discrete quantum group actions, and present some basic connections between FA-amenable actions and amenable discrete quantum groups. As an application, thinking of a state on a unital \(C^*\) -algebra as a “probability measure” on a compact quantum space, we show that amenability for a discrete quantum group is equivalent to both of FA-amenability for an action of this discrete quantum group on a compact quantum space and the existence of this kind of “probability measure” that is FA-invariant under this action. PubDate: 2022-09-13 DOI: 10.1007/s43037-022-00217-2

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Abstract: Abstract Consider a linear functional L defined on the space \({\mathcal {D}}[s]\) of Dirichlet polynomials with real coefficients and the set \({\mathcal {D}}_+[s]\) of non-negative elements in \({\mathcal {D}}[s].\) An analogue of the Riesz–Haviland theorem in this context asks: What are all \({\mathcal {D}}_+[s]\) -positive linear functionals L, which are moment functionals' Since the space \({\mathcal {D}}[s],\) when considered as a subspace of \(C([0, \infty ), {\mathbb {R}}),\) fails to be an adapted space in the sense of Choquet, the general form of Riesz–Haviland theorem is not applicable in this situation. In an attempt to answer the forgoing question, we arrive at the notion of a moment sequence, which we call the Hausdorff log-moment sequence. Apart from an analogue of the Riesz–Haviland theorem, we show that any Hausdorff log-moment sequence is a linear combination of \(\{1, 0, \ldots , \}\) and \(\{f(\log (n)\}_{n \geqslant 1}\) for a completely monotone function \(f : [0, \infty ) \rightarrow [0, \infty ).\) Moreover, such an f is uniquely determined by the sequence in question. PubDate: 2022-09-09 DOI: 10.1007/s43037-022-00208-3

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Abstract: Abstract It was shown in [Colloq. Math. 131(2), 219--231 (2013)] that one can extend the domain of Fourier transform of a commutative hypergroup K to \(L^p(K)\) for \(1\le p \le 2\) , and the Hausdorff–Young inequality holds true for these cases. In this article, we examine the structure of non-zero functions in \(L^p(K)\) for which equality is attained in the Hausdorff–Young inequality, for \(1<p<2\) , and further provide a characterization for the basic uncertainty principle for commutative hypergroups with non-trivial center. PubDate: 2022-09-01 DOI: 10.1007/s43037-022-00211-8

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Abstract: Abstract We consider in this paper the following interpolation problem: characterize the class of pairs (P, Q) of self-adjoint operators on a complex Hilbert space \({\mathcal {H}}\) satisfying \(CPC=Q\) for some conjugation C on \({\mathcal {H}}\) . Based on a concrete description of normal operators commuting with conjugations, we give a solution to the problem in the case that P commutes with Q. PubDate: 2022-08-29 DOI: 10.1007/s43037-022-00216-3