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1. CMB Online first

Awonusika, Richard; Taheri, Ali
 A spectral identity on Jacobi polynomials and its analytic implications The Jacobi coefficients $c^{\ell}_{j}(\alpha,\beta)$ ($1\leq j\leq \ell$, $\alpha,\beta\gt -1$) are linked to the Maclaurin spectral expansion of the Schwartz kernel of functions of the Laplacian on a compact rank one symmetric space. It is proved that these coefficients can be computed by transforming the even derivatives of the the Jacobi polynomials $P_{k}^{(\alpha,\beta)}$ ($k\geq 0, \alpha,\beta\gt -1$) into a spectral sum associated with the Jacobi operator. The first few coefficients are explicitly computed and a direct trace interpretation of the Maclaurin coefficients is presented. Keywords:Jacobi coefficient, Laplace-Beltrami operator, symmetric space, Maclaurin expansion, Jacobi polynomialCategories:33C05, 33C45, 35A08, 35C05, 35C10, 35C15

2. CMB Online first

Rocha, Pablo Alejandro
 A remark on certain integral operators of fractional type For $m, n \in \mathbb{N}$, $1\lt m \leq n$, we write $n = n_1 + \dots + n_m$ where $\{ n_1, \dots, n_m \} \subset \mathbb{N}$. Let $A_1, \dots, A_m$ be $n \times n$ singular real matrices such that $\bigoplus_{i=1}^{m} \bigcap_{1\leq j \neq i \leq m} \mathcal{N}_j = \mathbb{R}^{n},$ where $\mathcal{N}_j = \{ x : A_j x = 0 \}$, $dim(\mathcal{N}_j)=n-n_j$ and $A_1+ \dots+ A_m$ is invertible. In this paper we study integral operators of the form $T_{r}f(x)= \int_{\mathbb{R}^{n}} \, |x-A_1 y|^{-n_1 + \alpha_1} \cdots |x-A_m y|^{-n_m + \alpha_m} f(y) \, dy,$ $n_1 + \dots + n_m = n$, $\frac{\alpha_1}{n_1} = \dots = \frac{\alpha_m}{n_m}=r$, $0 \lt r \lt 1$, and the matrices $A_i$'s are as above. We obtain the $H^{p}(\mathbb{R}^{n})-L^{q}(\mathbb{R}^{n})$ boundedness of $T_r$ for $0\lt p\lt \frac{1}{r}$ and $\frac{1}{q}=\frac{1}{p} - r$. Keywords:integral operator, Hardy spaceCategories:42B20, 42B30

3. CMB Online first

Bénéteau, Catherine Anne; Fleeman, Matthew C.; Khavinson, Dmitry S.; Seco, Daniel; Sola, Alan
 Remarks on inner functions and optimal approximants We discuss the concept of inner function in reproducing kernel Hilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to $1/f$, where $f$ is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modified to produce inner functions. Keywords:inner function, reproducing Kernel Hilbert Space, operator-theoretic function theoryCategories:46E22, 30J05

4. CMB Online first

Jeong, Imsoon; de Dios Pérez, Juan; Suh, Young Jin; Woo, Changhwa
 Lie derivatives and Ricci tensor on real hypersurfaces in complex two-plane Grassmannians On a real hypersurface $M$ in a complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$ we have the Lie derivation ${\mathcal L}$ and a differential operator of order one associated to the generalized Tanaka-Webster connection $\widehat {\mathcal L} ^{(k)}$. We give a classification of real hypersurfaces $M$ on $G_2({\mathbb C}^{m+2})$ satisfying $\widehat {\mathcal L} ^{(k)}_{\xi}S={\mathcal L}_{\xi}S$, where $\xi$ is the Reeb vector field on $M$ and $S$ the Ricci tensor of $M$. Keywords:real hypersurface, complex two-plane Grassmannian, Hopf hypersurface, shape operator, Ricci tensor, Lie derivationCategories:53C40, 53C15

5. CMB 2017 (vol 60 pp. 462)

Bayart, Frédéric; Gauthier, Paul M
 Functions Universal for All Translation Operators in Several Complex Variables We prove the existence of a (in fact many) holomorphic function $f$ in $\mathbb{C}^d$ such that, for any $a\neq 0$, its translations $f(\cdot+na)$ are dense in $H(\mathbb{C}^d)$. Keywords:hypercyclic operator, translation operatorCategories:47A16, 32E20

6. CMB Online first

Saito, Hiroki; Tanaka, Hitoshi
 The Fefferman-Stein type inequalities for strong and directional maximal operators in the plane The Fefferman-Stein type inequalities for strong maximal operator and directional maximal operator are verified with an additional composition of the Hardy-Littlewood maximal operator in the plane. Keywords:directional maximal operator, Fefferman-Stein type inequality, Hardy-Littlewood maximal operator, strong maximal operatorCategories:42B25, 42B35

7. CMB 2017 (vol 60 pp. 747)

Huang, Yanhe; Sottile, Frank; Zelenko, Igor
 Injectivity of Generalized Wronski Maps We study linear projections on PlÃ¼cker space whose restriction to the Grassmannian is a non-trivial branched cover. When an automorphism of the Grassmannian preserves the fibers, we show that the Grassmannian is necessarily of $m$-dimensional linear subspaces in a symplectic vector space of dimension $2m$, and the linear map is the Lagrangian involution. The Wronski map for a self-adjoint linear differential operator and pole placement map for symmetric linear systems are natural examples. Keywords:Wronski map, PlÃ¼cker embedding, curves in Lagrangian Grassmannian, self-adjoint linear differential operator, symmetric linear control system, pole placement mapCategories:14M15, 34A30, 93B55

8. CMB 2016 (vol 60 pp. 196)

Rhaly, H. C.
 Corrigendum to "Generalized CesÃ ro Matrices" This note corrects an error in Theorem 1 of "Generalized CesÃ ro matrices" Canad. Math. Bull. 27 (1984), no. 4, 417-422. Keywords:Cesaro operator, Hilbert-Schmidt operator, numerical rangeCategories:47B99, 47A12, 47B10, 47B38

9. CMB 2016 (vol 60 pp. 411)

Stoyanov, Luchezar
 On Gibbs Measures and Spectra of Ruelle Transfer Operators We prove a comprehensive version of the Ruelle-Perron-Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is that the HÃ¶lder constant of the function generating the operator appears only polynomially, not exponentially as in previous known estimates. Keywords:subshift of finite type, Ruelle transfer operator, Gibbs measureCategories:37A05, 37B10

10. CMB 2016 (vol 60 pp. 131)

Gürbüz, Ferit
 Some Estimates for Generalized Commutators of Rough Fractional Maximal and Integral Operators on Generalized Weighted Morrey Spaces In this paper, we establish $BMO$ estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively. Keywords:fractional integral operator, fractional maximal operator, rough kernel, generalized commutator, $A(p,q)$ weight, generalized weighted Morrey spaceCategories:42B20, 42B25

11. CMB 2016 (vol 60 pp. 217)

Wang, Yuanyi
 Condition $C'_{\wedge}$ of Operator Spaces In this paper, we study condition $C'_{\wedge}$ which is a projective tensor product analogue of condition $C'$. We show that the finite-dimensional OLLP operator spaces have condition $C'_{\wedge}$ and $M_{n}$ $(n\gt 2)$ does not have that property. Keywords:operator space, local theory, tensor productCategory:46L07

12. CMB 2016 (vol 60 pp. 872)

Xu, Xu; Zhu, Laiyi
 Rational Function Operators from Poisson Integrals In this paper, we construct two classes of rational function operators by using the Poisson integrals of the function on the whole real axis. The convergence rates of the uniform and mean approximation of such rational function operators on the whole real axis are studied. Keywords:rational function operators, Poisson integrals, convergence rate, uniform approximation, mean approximationCategories:41A20, 41A25, 41A35

13. CMB 2016 (vol 60 pp. 586)

Liu, Feng; Wu, Huoxiong
 Endpoint Regularity of Multisublinear Fractional Maximal Functions In this paper we investigate the endpoint regularity properties of the multisublinear fractional maximal operators, which include the multisublinear Hardy-Littlewood maximal operator. We obtain some new bounds for the derivative of the one-dimensional multisublinear fractional maximal operators acting on vector-valued function $\vec{f}=(f_1,\dots,f_m)$ with all $f_j$ being $BV$-functions. Keywords:multisublinear fractional maximal operators, Sobolev spaces, bounded variationCategories:42B25, 46E35

14. CMB 2016 (vol 59 pp. 813)

Kaimakamis, George; Panagiotidou, Konstantina; Pérez, Juan de Dios
 A Classification of Three-dimensional Real Hypersurfaces in Non-flat Complex Space Forms in Terms of Their generalized Tanaka-Webster Lie Derivative On a real hypersurface $M$ in a non-flat complex space form there exist the Levi-Civita and the k-th generalized Tanaka-Webster connections. The aim of the present paper is to study three dimensional real hypersurfaces in non-flat complex space forms, whose Lie derivative of the structure Jacobi operator with respect to the Levi-Civita connections coincides with the Lie derivative of it with respect to the k-th generalized Tanaka-Webster connection. The Lie derivatives are considered in direction of the structure vector field and in directions of any vecro field orthogonal to the structure vector field. Keywords:$k$-th generalized Tanaka-Webster connection, non-flat complex space form, real hypersurface, Lie derivative, structure Jacobi operatorCategories:53C15, 53B25

15. CMB 2016 (vol 60 pp. 77)

Christ, Michael; Rieffel, Marc A.
 Nilpotent Group C*-algebras as Compact Quantum Metric Spaces Let $\mathbb{L}$ be a length function on a group $G$, and let $M_\mathbb{L}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on $\lt(G)$. Following Connes, $M_\mathbb{L}$ can be used as a Dirac'' operator for the reduced group C*-algebra $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a compact quantum metric space''). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups. Keywords:group C*-algebra, Dirac operator, quantum metric space, discrete nilpotent group, polynomial growthCategories:46L87, 20F65, 22D15, 53C23, 58B34

16. CMB 2016 (vol 59 pp. 564)

Li, Boyu
 Normal Extensions of Representations of Abelian Semigroups A commuting family of subnormal operators need not have a commuting normal extension. We study when a representation on an abelian semigroup can be extended to a normal representation, and show that it suffices to extend the set of generators to commuting normals. We also extend a result due to Athavale to representations on abelian lattice ordered semigroups. Keywords:subnormal operator, normal extension, regular dilation, lattice ordered semigroupCategories:47B20, 47A20, 47D03

17. CMB 2016 (vol 59 pp. 734)

Dimassi, Mouez
 Semi-classical Asymptotics for SchrÃ¶dinger Operator with Oscillating Decaying Potential We study the distribution of the discrete spectrum of the SchrÃ¶dinger operator perturbed by a fast oscillating decaying potential depending on a small parameter $h$. Keywords:periodic SchrÃ¶dinger operator, semi-classical asymptotics, effective Hamiltonian, asymptotic expansion, spectral shift functionCategories:81Q10, 35P20, 47A55, 47N50, 81Q15

18. CMB 2016 (vol 59 pp. 497)

De Carli, Laura; Samad, Gohin Shaikh
 One-parameter Groups of Operators and Discrete Hilbert Transforms We show that the discrete Hilbert transform and the discrete Kak-Hilbert transform are infinitesimal generator of one-parameter groups of operators in $\ell^2$. Keywords:discrete Hilbert transform, groups of operators, isometriesCategories:42A45, 42A50, 41A44

19. CMB 2016 (vol 59 pp. 417)

Song, Hongxue; Chen, Caisheng; Yan, Qinglun
 Existence of Multiple Solutions for a $p$-Laplacian System in $\textbf{R}^{N}$ with Sign-changing Weight Functions In this paper, we consider the quasi-linear elliptic problem \left\{ \begin{aligned} & -M \left(\int_{\mathbb{R}^{N}}|x|^{-ap}|\nabla u|^{p}dx \right){\rm div} \left(|x|^{-ap}|\nabla u|^{p-2}\nabla u \right) \\ & \qquad=\frac{\alpha}{\alpha+\beta}H(x)|u|^{\alpha-2}u|v|^{\beta}+\lambda h_{1}(x)|u|^{q-2}u, \\ & -M \left(\int_{\mathbb{R}^{N}}|x|^{-ap}|\nabla v|^{p}dx \right){\rm div} \left(|x|^{-ap}|\nabla v|^{p-2}\nabla v \right) \\ & \qquad=\frac{\beta}{\alpha+\beta}H(x)|v|^{\beta-2}v|u|^{\alpha}+\mu h_{2}(x)|v|^{q-2}v, \\ &u(x)\gt 0,\quad v(x)\gt 0, \quad x\in \mathbb{R}^{N} \end{aligned} \right. where $\lambda, \mu\gt 0$, $1\lt p\lt N$, $1\lt q\lt p\lt p(\tau+1)\lt \alpha+\beta\lt p^{*}=\frac{Np}{N-p}$, $0\leq a\lt \frac{N-p}{p}$, $a\leq b\lt a+1$, $d=a+1-b\gt 0$, $M(s)=k+l s^{\tau}$, $k\gt 0$, $l, \tau\geq0$ and the weight $H(x), h_{1}(x), h_{2}(x)$ are continuous functions which change sign in $\mathbb{R}^{N}$. We will prove that the problem has at least two positive solutions by using the Nehari manifold and the fibering maps associated with the Euler functional for this problem. Keywords:Nehari manifold, quasilinear elliptic system, $p$-Laplacian operator, concave and convex nonlinearitiesCategory:35J66

20. CMB 2016 (vol 59 pp. 693)

Chen, Chung-Chuan
 Recurrence of Cosine Operator Functions on Groups In this note, we study the recurrence and topologically multiple recurrence of a sequence of operators on Banach spaces. In particular, we give a sufficient and necessary condition for a cosine operator function, induced by a sequence of operators on the Lebesgue space of a locally compact group, to be topologically multiply recurrent. Keywords:topologically multiple recurrence, recurrence, topological transitivity, hypercyclicity, cosine operator functionCategories:47A16, 54B20, 43A15

21. CMB 2016 (vol 59 pp. 326)

Jiang, Chunlan; Shi, Rui
 On the Uniqueness of Jordan Canonical Form Decompositions of Operators by $K$-theoretical Data In this paper, we develop a generalized Jordan canonical form theorem for a certain class of operators in $\mathcal {L}(\mathcal {H})$. A complete criterion for similarity for this class of operators in terms of $K$-theory for Banach algebras is given. Keywords:strongly irreducible operator, similarity invariant, reduction theory of von Neumann algebras, $K$-theoryCategories:47A15, 47C15, 47A65

22. CMB 2016 (vol 59 pp. 354)

Li, Chi-Kwong; Tsai, Ming-Cheng
 Factoring a Quadratic Operator as a Product of Two Positive Contractions Let $T$ be a quadratic operator on a complex Hilbert space $H$. We show that $T$ can be written as a product of two positive contractions if and only if $T$ is of the form \begin{equation*} aI \oplus bI \oplus \begin{pmatrix} aI & P \cr 0 & bI \cr \end{pmatrix} \quad \text{on} \quad H_1\oplus H_2\oplus (H_3\oplus H_3) \end{equation*} for some $a, b\in [0,1]$ and strictly positive operator $P$ with $\|P\| \le |\sqrt{a} - \sqrt{b}|\sqrt{(1-a)(1-b)}.$ Also, we give a necessary condition for a bounded linear operator $T$ with operator matrix $\big( \begin{smallmatrix} T_1 & T_3 \\ 0 & T_2\cr \end{smallmatrix} \big)$ on $H\oplus K$ that can be written as a product of two positive contractions. Keywords:quadratic operator, positive contraction, spectral theoremCategories:47A60, 47A68, 47A63

23. CMB 2015 (vol 58 pp. 723)

Castro, Alfonso; Fischer, Emily
 Infinitely Many Rotationally Symmetric Solutions to a Class of Semilinear Laplace-Beltrami Equations on Spheres We show that a class of semilinear Laplace-Beltrami equations on the unit sphere in $\mathbb{R}^n$ has infinitely many rotationally symmetric solutions. The solutions to these equations are the solutions to a two point boundary value problem for a singular ordinary differential equation. We prove the existence of such solutions using energy and phase plane analysis. We derive a Pohozaev-type identity in order to prove that the energy to an associated initial value problem tends to infinity as the energy at the singularity tends to infinity. The nonlinearity is allowed to grow as fast as $|s|^{p-1}s$ for $|s|$ large with $1 \lt p \lt (n+5)/(n-3)$. Keywords:Laplace-Beltrami operator, semilinear equation, rotational solution, superlinear nonlinearity, sub-super critical nonlinearityCategories:58J05, 35A24

24. CMB 2015 (vol 58 pp. 486)

Duc, Dinh Thanh; Nhan, Nguyen Du Vi; Xuan, Nguyen Tong
 Inequalities for Partial Derivatives and their Applications We present various weighted integral inequalities for partial derivatives acting on products and compositions of functions which are applied to establish some new Opial-type inequalities involving functions of several independent variables. We also demonstrate the usefulness of our results in the field of partial differential equations. Keywords:inequality for integral, Opial-type inequality, HÃ¶lder's inequality, partial differential operator, partial differential equationCategories:26D10, 35A23

25. CMB 2015 (vol 58 pp. 808)

Liu, Feng; Wu, Huoxiong
 On the Regularity of the Multisublinear Maximal Functions This paper is concerned with the study of the regularity for the multisublinear maximal operator. It is proved that the multisublinear maximal operator is bounded on first-order Sobolev spaces. Moreover, two key point-wise inequalities for the partial derivatives of the multisublinear maximal functions are established. As an application, the quasi-continuity on the multisublinear maximal function is also obtained. Keywords:regularity, multisublinear maximal operator, Sobolev spaces, partial deviative, quasicontinuityCategories:42B25, 46E35
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