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1.
We study set-gradings on proper \(H^*\)-algebras A, which are compatible with the involution and the inner product of A, that will be called orthogonal-gradings. If A is an arbitrary \(H^*\)-algebra with a fine grading, we obtain a (fine) orthogonal-graded version of the main structure theorem for proper arbitrary \(H^*\)-algebras. If A is associative, we show that any fine orthogonal-grading is either a group-grading or a (non-group grading) Peirce decomposition of A respect to a family of orthogonal projections. If A is alternative, we prove that any fine orthogonal-grading is either a fine orthogonal-grading of a (proper) associative \(H^*\)-algebra, or a \({\mathbb Z}_2^3\)-grading of the complex octonions \({\mathbb O}\) or a non-group grading which is a refinement of the Peirce decomposition of \({\mathbb O}\) respect to its family of orthogonal projections. Finally, we also show that any orthogonal-grading on the real octonion division algebra is necessarily a group-grading.  相似文献   

2.
The dynamics of functions \(f_\lambda (z)= \lambda \frac{\mathrm{e}^{z}}{z+1}\ \text{ for }\ z\in \mathbb {C}, \lambda >0\) is studied showing that there exists \(\lambda ^* > 0\) such that the Julia set of \(f_\lambda \) is disconnected for \(0< \lambda < \lambda ^*\) whereas it is the whole Riemann sphere for \(\lambda > \lambda ^*\). Further, for \(0< \lambda < \lambda ^*\), the Julia set is a disjoint union of two topologically and dynamically distinct completely invariant subsets, one of which is totally disconnected. The union of the escaping set and the backward orbit of \(\infty \) is shown to be disconnected for \(0<\lambda < \lambda ^*\) whereas it is connected for \(\lambda > \lambda ^*\). For complex \(\lambda \), it is proved that either all multiply connected Fatou components ultimately land on an attracting or parabolic domain containing the omitted value of the function or the Julia set is connected. In the latter case, the Fatou set can be empty or consists of Siegel disks. All these possibilities are shown to occur for suitable parameters. Meromorphic functions \(E_n(z) =\mathrm{e}^{z}(1+z+\frac{z^2}{2!}+\cdots +\frac{z^n}{n!})^{-1}\), which we call exponential-like, are studied as a generalization of \(f(z)=\frac{\mathrm{e}^{z}}{z+1}\) which is nothing but \(E_1(z)\). This name is justified by showing that \(E_n\) has an omitted value 0 and there are no other finite singular value. In fact, it is shown that there is only one singularity over 0 as well as over \(\infty \) and both are direct. Non-existence of Herman rings are proved for \(\lambda E_n \).  相似文献   

3.
Let \(\varphi \) be an arbitrary linear-fractional self-map of the unit disk \({\mathbb {D}}\) and consider the composition operator \(C_{-1, \varphi }\) and the Toeplitz operator \(T_{-1,z}\) on the Hardy space \(H^2\) and the corresponding operators \(C_{\alpha , \varphi }\) and \(T_{\alpha , z}\) on the weighted Bergman spaces \(A^2_{\alpha }\) for \(\alpha >-1\). We prove that the unital C\(^*\)-algebra \(C^*(T_{\alpha , z}, C_{\alpha , \varphi })\) generated by \(T_{\alpha , z}\) and \(C_{\alpha , \varphi }\) is unitarily equivalent to \(C^*(T_{-1, z}, C_{-1, \varphi }),\) which extends a known result for automorphism-induced composition operators. For maps \(\varphi \) that are not automorphisms of \({\mathbb {D}}\), we show that \(C^*(C_{\alpha , \varphi }, {\mathcal {K}}_{\alpha })\) is unitarily equivalent to \(C^*(C_{-1, \varphi }, {\mathcal {K}}_{-1})\), where \({\mathcal {K}}_{\alpha }\) and \({\mathcal {K}}_{-1}\) denote the ideals of compact operators on \(A^2_{\alpha }\) and \(H^2\), respectively, and apply existing structure theorems for \(C^*(C_{-1, \varphi }, {\mathcal {K}}_{-1})/{\mathcal {K}}_{-1}\) to describe the structure of \(C^*(C_{\alpha , \varphi }, {\mathcal {K}}_{\alpha })/\mathcal {K_{\alpha }}\), up to isomorphism. We also establish a unitary equivalence between related weighted composition operators induced by maps \(\varphi \) that fix a point on the unit circle.  相似文献   

4.
An operator \(S_{\varphi ,\psi }^{u}\in \mathcal {L}(L^2)\) is called the dilation of a truncated Toeplitz operator if for two symbols \(\varphi ,\psi \in L^{\infty }\) and an inner function u,
$$\begin{aligned} S_{\varphi ,\psi }^{u}f=\varphi P_uf+\psi Q_uf \end{aligned}$$
holds for \(f\in {L}^{2}\) where \(P_{u}\) denotes the orthogonal projection of \(L^2\) onto the model space \(\mathcal { K}_{u}^2=H^2{\ominus }{{u}H^2}\) and \(Q_u=I-P_u.\) In this paper, we study properties of the dilation of truncated Toeplitz operators on \(L^{2}\). In particular, we provide conditions for the dilation of truncated Toeplitz operators to be normal. As some applications, we give several examples of such operators.
  相似文献   

5.
We study the following elliptic problem \(-A(u) = \lambda u^q\) with Dirichlet boundary conditions, where \(A(u) (x) = \Delta u (x) \chi _{D_1} (x)+ \Delta _p u(x) \chi _{D_2}(x)\) is the Laplacian in one part of the domain, \(D_1\), and the p-Laplacian (with \(p>2\)) in the rest of the domain, \(D_2 \). We show that this problem exhibits a concave–convex nature for \(1<q<p-1\). In fact, we prove that there exists a positive value \(\lambda ^*\) such that the problem has no positive solution for \(\lambda > \lambda ^*\) and a minimal positive solution for \(0<\lambda < \lambda ^*\). If in addition we assume that p is subcritical, that is, \(p<2N/(N-2)\) then there are at least two positive solutions for almost every \(0<\lambda < \lambda ^*\), the first one (that exists for all \(0<\lambda < \lambda ^*\)) is obtained minimizing a suitable functional and the second one (that is proven to exist for almost every \(0<\lambda < \lambda ^*\)) comes from an appropriate (and delicate) mountain pass argument.  相似文献   

6.
Let G be a locally compact abelian group, \(\omega \) be a weighted function on \({\mathbb {R}}^+\), and let \(\mathfrak {D}\) be the Banach algebra \(L_0^\infty (G)^*\) or \(L_0^\infty (\omega )^*\). In this paper, we investigate generalized derivations on the noncommutative Banach algebra \(\mathfrak {D}\). We characterize \(\textsf {k}\)-(skew) centralizing generalized derivations of \(\mathfrak {D}\) and show that the zero map is the only \(\textsf {k}\)-skew commuting generalized derivation of \(\mathfrak {D}\). We also investigate the Singer–Wermer conjecture for generalized derivations of \(\mathfrak {D}\) and prove that the Singer–Wermer conjecture holds for a generalized derivation of \(\mathfrak {D}\) if and only if it is a derivation; or equivalently, it is nilpotent. Finally, we investigate the orthogonality of generalized derivations of \(L_0^\infty (\omega )^*\) and give several necessary and sufficient conditions for orthogonal generalized derivations of \(L_0^\infty (\omega )^*\).  相似文献   

7.
We solve a characterization problem for dual Hardy-space Toeplitz operators on the unit sphere \({\mathbb{S}_{n}}\) in \({\mathbb{C}^{n}}\) posed by Guediri (Acta Math Sin (English series) 29(9):1791–1808, 2013). Our proof relies on the observation that dual Toeplitz operators on the orthogonal complement \({H^{2}(\mathbb{S}_{n})^{\bot}}\) of the Hardy space in L 2 can be viewed as Toeplitz operators with respect to a suitable spherical isometry. This correspondence also allows us to determine the commutator ideal of the dual Toeplitz C *-algebra.  相似文献   

8.
In this paper we show that the C*-algebra generated by radial Toeplitz operators with \(L_{\infty }\)-symbols acting on the Fock space is isometrically isomorphic to the C*-algebra of bounded sequences uniformly continuous with respect to the square-root-metric \(\rho (j,k)=|\sqrt{j}-\sqrt{k}\,|\). More precisely, we prove that the sequences of eigenvalues of radial Toeplitz operators form a dense subset of the latter C*-algebra of sequences.  相似文献   

9.
We study the external and internal Zappa–Szép product of topological groupoids. We show that under natural continuity assumptions the Zappa–Szép product groupoid is étale if and only if the individual groupoids are étale. In our main result we show that the \(C^*\)-algebra of a locally compact Hausdorff étale Zappa–Szép product groupoid is a \(C^*\)-blend, in the sense of Exel, of the individual groupoid \(C^*\)-algebras. We finish with some examples, including groupoids built from \(*\)-commuting endomorphisms, and skew product groupoids.  相似文献   

10.
In an extension of Kendall’s \(\tau \), Bergsma and Dassios (Bernoulli 20(2):1006–1028, 2014) introduced a covariance measure \(\tau ^*\) for two ordinal random variables that vanishes if and only if the two variables are independent. For a sample of size n, a direct computation of \(t^*\), the empirical version of \(\tau ^*\), requires \(O(n^4)\) operations. We derive an algorithm that computes the statistic using only \(O \left( n^2\log (n)\right) \) operations.  相似文献   

11.
We present dimension-free reverse Hölder inequalities for strong \(A^*_p\) weights, \(1\le p < \infty \). We also provide a proof for the full range of local integrability of \(A_1^*\) weights. The common ingredient is a multidimensional version of Riesz’s “rising sun” lemma. Our results are valid for any nonnegative Radon measure with no atoms. For \(p=\infty \), we also provide a reverse Hölder inequality for certain product measures. As a corollary we derive mixed \(A_p^*-A_\infty ^*\) weighted estimates.  相似文献   

12.
Representations on Hilbert spaces for a nonlocal C*-algebra \({{\mathfrak {B}}}\) of singular integral operators with piecewise slowly oscillating coefficients and unitary shift operators are constructed. The group of unitary shift operators U g of the C*-algebra \({{\mathfrak {B}}}\) is associated with an amenable discrete group of homeomorphisms \({g:{\mathbb{T}}\to{\mathbb{T}}}\) that have piecewise continuous derivatives and the same nonempty set of periodic points. An isometric C*-algebra homomorphism of the quotient C*-algebra \({{\mathfrak {B}}^\pi={\mathfrak {B}}/{\mathcal {K}}}\), where \({{\mathcal {K}}}\) is the ideal of compact operators, into an n × n matrix algebra associated to a C*-algebra \({{\mathfrak {B}}_0}\) of singular integral operators with shifts having only fixed points is established making use of a spectral measure. Based on this generalization of the Litvinchuk–Gohberg–Krupnik reduction scheme, a symbol calculus for the C*-algebra \({{\mathfrak {B}}}\) as well as a Fredholm criterion for the operators in \({{\mathfrak {B}}}\) are obtained.  相似文献   

13.
In this paper, we establish the limit of empirical spectral distributions of quaternion sample covariance matrices. Motivated by Bai and Silverstein (Spectral analysis of large dimensional random matrices, Springer, New York, 2010) and Mar?enko and Pastur (Matematicheskii Sb, 114:507–536, 1967), we can extend the results of the real or complex sample covariance matrix to the quaternion case. Suppose \(\mathbf X_n = ({x_{jk}^{(n)}})_{p\times n}\) is a quaternion random matrix. For each \(n\), the entries \(\{x_{ij}^{(n)}\}\) are independent random quaternion variables with a common mean \(\mu \) and variance \(\sigma ^2>0\). It is shown that the empirical spectral distribution of the quaternion sample covariance matrix \(\mathbf S_n=n^{-1}\mathbf X_n\mathbf X_n^*\) converges to the Mar?enko–Pastur law as \(p\rightarrow \infty \), \(n\rightarrow \infty \) and \(p/n\rightarrow y\in (0,+\infty )\).  相似文献   

14.
15.
Let X and \(X^*\) denote a restricted ray transform along curves and a corresponding backprojection operator, respectively. Theoretical analysis of reconstruction from the data Xf is usually based on a study of the composition \(X^* D X\), where D is some local operator (usually a derivative). If \(X^*\) is chosen appropriately, then \(X^* D X\) is a Fourier integral operator (FIO) with singular symbol. The singularity of the symbol leads to the appearance of artifacts (added singularities) that can be as strong as the original (or, useful) singularities. By choosing D in a special way one can reduce the strength of added singularities, but it is impossible to get rid of them completely. In the paper we follow a similar approach, but make two changes. First, we replace D with a nonlocal operator \(\tilde{D}\) that integrates Xf along a curve in the data space. The result \(\tilde{D} Xf\) resembles the generalized Radon transform R of f. The function \(\tilde{D} Xf\) is defined on pairs \((x_0,\Theta )\in U\times S^2\), where \(U\subset {\mathbb R}^3\) is an open set containing the support of f, and \(S^2\) is the unit sphere in \({\mathbb R}^3\). Second, we replace \(X^*\) with a backprojection operator \(R^*\) that integrates with respect to \(\Theta \) over \(S^2\). It turns out that if \(\tilde{D}\) and \(R^*\) are appropriately selected, then the composition \(R^* \tilde{D} X\) is an elliptic pseudodifferential operator of order zero with principal symbol 1. Thus, we obtain an approximate reconstruction formula that recovers all the singularities correctly and does not produce artifacts. The advantage of our approach is that by inserting \(\tilde{D}\) we get access to the frequency variable \(\Theta \). In particular, we can incorporate suitable cut-offs in \(R^*\) to eliminate bad directions \(\Theta \), which lead to added singularities.  相似文献   

16.
We construct an uncountable family of periodic locally soluble groups which are hereditarily just infinite. We also show that the associated full \(C^*\)-algebra \(C^*(G)\) is just infinite for many groups G in this family.  相似文献   

17.
Let H be a Hopf algebra. Any finite-dimensional lifting of \(V\in {}^{H}_{H}\mathcal {YD}\) arising as a cocycle deformation of \(A={\mathfrak {B}}(V)\#H\) defines a twist in the Hopf algebra \(A^*\), via dualization. We follow this recipe to write down explicit examples and show that it extends known techniques for defining twists. We also contribute with a detailed survey about twists in braided categories.  相似文献   

18.
We study the behaviour of sequences \(U_2^n X U_1^{-n}\), where \(U_1, U_2\) are unitary operators, whose spectral measures are singular with respect to the Lebesgue measure, and the commutator \(XU_1-U_2X\) is small in a sense. The conjecture about the weak averaged convergence of the difference \(U_2^n X U_1^{-n}-U_2^{-n} X U_1^n\) to the zero operator is discussed and its connection with complex-symmetric operators is established in a general situation. For a model case where \(U_1=U_2\) is the unitary operator of multiplication by z on \(L^2(\mu )\), sufficient conditions for the convergence as in the Conjecture are given in terms of kernels of integral operators.  相似文献   

19.
R. M. Causey 《Positivity》2017,21(4):1507-1525
We prove the optimal estimate between the Szlenk and \(w^*\)-dentability indices of an arbitrary \(w^*\)-compact subset of the dual of a Banach space. For a given \(w^*\)-compact, convex subset K of the dual of a Banach space, we introduce a two player game the winning strategies of which determine the Szlenk index of K. We give applications to the \(w^*\)-dentability index of a Banach space and of an operator.  相似文献   

20.
The main aim of this paper is to discuss the third Hankel determinants for three classes: \(S^*\) of starlike functions, \(\mathcal {K}\) of convex functions and \(\mathcal {R}\) of functions whose derivative has a positive real part. Moreover, the sharp results for twofold and threefold symmetric functions from these classes are obtained.  相似文献   

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