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1. CMB Online first

Bu, Shangquan; Cai, Gang
 Periodic solutions of second order degenerate differential equations with delay in Banach spaces We give necessary and sufficient conditions of the $L^p$-well-posedness (resp. $B_{p,q}^s$-well-posedness) for the second order degenerate differential equation with finite delays: $(Mu)''(t)+Bu'(t)+Au(t)=Gu'_t+Fu_t+f(t),(t\in [0,2\pi])$ with periodic boundary conditions $(Mu)(0)=(Mu)(2\pi)$, $(Mu)'(0)=(Mu)'(2\pi)$, where $A, B, M$ are closed linear operators on a complex Banach space $X$ satisfying $D(A)\cap D(B)\subset D(M)$, $F$ and $G$ are bounded linear operators from $L^p([-2\pi,0];X)$ (resp. $B_{p,q}^s([-2\pi,0];X)$) into $X$. Keywords:second order degenerate differential equation, Fourier multiplier theorem, well-posedness, Lebesgue-Bochner space, Besov spaceCategories:34G10, 34K30, 43A15, 47D06

2. 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

3. CMB 2017 (vol 60 pp. 536)

Kalaj, David; Vujadinović, Djordjije
 The Gradient of a Solution of the Poisson Equation in the Unit Ball and Related Operators In this paper we determine the $L^1\to L^1$ and $L^{\infty}\to L^\infty$ norms of an integral operator $\mathcal{N}$ related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions. Keywords:MÃ¶bius transformation, Poisson equation, Newtonian potential, Cauchy transform, Bessel functionCategories:35J05, 47G10

4. CMB Online first

Bu, Shangquan; Cai, Gang
 HÃ¶lder continuous solutions of degenerate differential equations with finite delay Using known operator-valued Fourier multiplier results on vector-valued HÃ¶lder continuous function spaces $C^\alpha (\mathbb R; X)$, we completely characterize the $C^\alpha$-well-posedness of the first order degenerate differential equations with finite delay $(Mu)'(t) = Au(t) + Fu_t + f(t)$ for $t\in\mathbb R$ by the boundedness of the $(M, F)$-resolvent of $A$ under suitable assumption on the delay operator $F$, where $A, M$ are closed linear operators on a Banach space $X$ satisfying $D(A)\cap D(M) \not=\{0\}$, the delay operator $F$ is a bounded linear operator from $C([-r, 0]; X)$ to $X$ and $r \gt 0$ is fixed. Keywords:well-posedness, degenerate differential equation, $\dot{C}^\alpha$-multiplier, HÃ¶lder continuous function spaceCategories:34N05, 34G10, 47D06, 47A10, 34K30

5. CMB 2017 (vol 60 pp. 561)

Kurdyka, Krzysztof; Paunescu, Laurentiu
 Nuij Type Pencils of Hyperbolic Polynomials Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic (i.e. has only real roots) then $p+sp'$ is also hyperbolic for any $s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials of the form $p_a(z,s): =p(z) +\sum_{k=1}^d a_ks^kp^{(k)}(z)$. We give a full characterization of those $a= (a_1, \dots, a_d) \in \mathbb{R}^d$ for which $p_a(z,s)$ is a pencil of hyperbolic polynomials. We give also a full characterization of those $a= (a_1, \dots, a_d) \in \mathbb{R}^d$ for which the associated families $p_a(z,s)$ admit universal determinantal representations. In fact we show that all these sequences come from special symmetric Toeplitz matrices. Keywords:hyperbolic polynomial, stable polynomial, determinantal representa- tion, symmetric Toeplitz matrixCategories:15A15, 30C10, 47A56

6. 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

7. CMB 2016 (vol 60 pp. 712)

Chen, Chung-Chuan
 Disjoint Hypercyclicity and Weighted Translations on Discrete Groups Let $1\leq p\lt \infty$, and let $G$ be a discrete group. We give a sufficient and necessary condition for weighted translation operators on the Lebesgue space $\ell^p(G)$ to be densely disjoint hypercyclic. The characterization for the dual of a weighted translation to be densely disjoint hypercyclic is also obtained. Keywords:disjoint hypercyclicity, topological transitivity, weighted translation, $\ell^p$-spaceCategories:47A16, 47B38, 43A15

8. CMB 2016 (vol 59 pp. 673)

Bačák, Miroslav; Kovalev, Leonid V.
 Lipschitz Retractions in Hadamard Spaces Via Gradient Flow Semigroups Let $X(n),$ for $n\in\mathbb{N},$ be the set of all subsets of a metric space $(X,d)$ of cardinality at most $n.$ The set $X(n)$ equipped with the Hausdorff metric is called a finite subset space. In this paper we are concerned with the existence of Lipschitz retractions $r\colon X(n)\to X(n-1)$ for $n\ge2.$ It is known that such retractions do not exist if $X$ is the one-dimensional sphere. On the other hand L. Kovalev has recently established their existence in case $X$ is a Hilbert space and he also posed a question as to whether or not such Lipschitz retractions exist for $X$ being a Hadamard space. In the present paper we answer this question in the positive. Keywords:finite subset space, gradient flow, Hadamard space, Lie-Trotter-Kato formula, Lipschitz retractionCategories:53C23, 47H20, 54E40, 58D07

9. 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

10. 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

11. CMB 2016 (vol 59 pp. 878)

Wang, Jianfei
 The Carleson Measure Problem Between Analytic Morrey Spaces The purpose of this paper is to characterize positive measure $\mu$ on the unit disk such that the analytic Morrey space $\mathcal{AL}_{p,\eta}$ is boundedly and compactly embedded to the tent space $\mathcal{T}_{q,1-\frac{q}{p}(1-\eta)}^{\infty}(\mu)$ for the case $1\leq q\leq p\lt \infty$ respectively. As an application, these results are used to establish the boundedness and compactness of integral operators and multipliers between analytic Morrey spaces. Keywords:Morrey space, Carleson measure problem, boundedness, compactnessCategories:30H35, 28A12, 47B38, 46E15

12. 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

13. CMB 2016 (vol 59 pp. 585)

Lin, Minghua
 A Determinantal Inequality Involving Partial Traces Let $\mathbf{A}$ be a density matrix in $\mathbb{M}_m\otimes \mathbb{M}_n$. Audenaert [J. Math. Phys. 48 (2007) 083507] proved an inequality for Schatten $p$-norms: $1+\|\mathbf{A}\|_p\ge \|\tr_1 \mathbf{A}\|_p+\|\tr_2 \mathbf{A}\|_p,$ where $\tr_1, \tr_2$ stand for the first and second partial trace, respectively. As an analogue of his result, we prove a determinantal inequality $1+\det \mathbf{A}\ge \det(\tr_1 \mathbf{A})^m+\det(\tr_2 \mathbf{A})^n.$ Keywords:determinantal inequality, partial trace, block matrixCategories:47B65, 15A45, 15A60

14. 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

15. 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

16. CMB 2015 (vol 59 pp. 3)

Alfuraidan, Monther Rashed
 The Contraction Principle for Multivalued Mappings on a Modular Metric Space with a Graph We study the existence of fixed points for contraction multivalued mappings in modular metric spaces endowed with a graph. The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. This paper can be seen as a generalization of Nadler's and Edelstein's fixed point theorems to modular metric spaces endowed with a graph. Keywords:fixed point theory, modular metric spaces, multivalued contraction mapping, connected digraph.Categories:47H09, 46B20, 47H10, 47E10

17. CMB 2015 (vol 58 pp. 402)

Tikuisis, Aaron Peter; Toms, Andrew
 On the Structure of Cuntz Semigroups in (Possibly) Nonunital C*-algebras We examine the ranks of operators in semi-finite $\mathrm{C}^*$-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple $\mathrm{C}^*$-algebra whose extreme tracial boundary is nonempty and finite contains positive operators of every possible rank, independent of the property of strict comparison. We then turn to nonunital simple algebras and establish criteria that imply that the Cuntz semigroup is recovered functorially from the Murray-von Neumann semigroup and the space of densely defined lower semicontinuous traces. Finally, we prove that these criteria are satisfied by not-necessarily-unital approximately subhomogeneous algebras of slow dimension growth. Combined with results of the first-named author, this shows that slow dimension growth coincides with $\mathcal Z$-stability, for approximately subhomogeneous algebras. Keywords:nuclear C*-algebras, Cuntz semigroup, dimension functions, stably projectionless C*-algebras, approximately subhomogeneous C*-algebras, slow dimension growthCategories:46L35, 46L05, 46L80, 47L40, 46L85

18. CMB 2015 (vol 58 pp. 241)

Botelho, Fernanda
 Isometries and Hermitian Operators on Zygmund Spaces In this paper we characterize the isometries of subspaces of the little Zygmund space. We show that the isometries of these spaces are surjective and represented as integral operators. We also show that all hermitian operators on these settings are bounded. Keywords:Zygmund spaces, the little Zygmund space, Hermitian operators, surjective linear isometries, generators of one-parameter groups of surjective isometriesCategories:46E15, 47B15, 47B38

19. CMB 2014 (vol 58 pp. 91)

Hasegawa, Kei
 Essential Commutants of Semicrossed Products Let $\alpha\colon G\curvearrowright M$ be a spatial action of countable abelian group on a "spatial" von Neumann algebra $M$ and $S$ be its unital subsemigroup with $G=S^{-1}S$. We explicitly compute the essential commutant and the essential fixed-points, modulo the Schatten $p$-class or the compact operators, of the w$^*$-semicrossed product of $M$ by $S$ when $M'$ contains no non-zero compact operators. We also prove a weaker result when $M$ is a von Neumann algebra on a finite dimensional Hilbert space and $(G,S)=(\mathbb{Z},\mathbb{Z}_+)$, which extends a famous result due to Davidson (1977) for the classical analytic Toeplitz operators. Keywords:essential commutant, semicrossed productCategories:47L65, 47A55

20. CMB 2014 (vol 58 pp. 276)

Johnson, William; Nasseri, Amir Bahman; Schechtman, Gideon; Tkocz, Tomasz
 Injective Tauberian Operators on $L_1$ and Operators with Dense Range on $\ell_\infty$ There exist injective Tauberian operators on $L_1(0,1)$ that have dense, nonclosed range. This gives injective, nonsurjective operators on $\ell_\infty$ that have dense range. Consequently, there are two quasi-complementary, noncomplementary subspaces of $\ell_\infty$ that are isometric to $\ell_\infty$. Keywords:$L_1$, Tauberian operator, $\ell_\infty$Categories:46E30, 46B08, 47A53

21. CMB 2014 (vol 58 pp. 9)

Chavan, Sameer
 Irreducible Tuples Without the Boundary Property We examine spectral behavior of irreducible tuples which do not admit boundary property. In particular, we prove under some mild assumption that the spectral radius of such an $m$-tuple $(T_1, \dots, T_m)$ must be the operator norm of $T^*_1T_1 + \cdots + T^*_mT_m$. We use this simple observation to ensure boundary property for an irreducible, essentially normal joint $q$-isometry provided it is not a joint isometry. We further exhibit a family of reproducing Hilbert $\mathbb{C}[z_1, \dots, z_m]$-modules (of which the Drury-Arveson Hilbert module is a prototype) with the property that any two nested unitarily equivalent submodules are indeed equal. Keywords:boundary representations, subnormal, joint p-isometryCategories:47A13, 46E22

22. CMB 2014 (vol 58 pp. 207)

Moslehian, Mohammad Sal; Zamani, Ali
 Exact and Approximate Operator Parallelism Extending the notion of parallelism we introduce the concept of approximate parallelism in normed spaces and then substantially restrict ourselves to the setting of Hilbert space operators endowed with the operator norm. We present several characterizations of the exact and approximate operator parallelism in the algebra $\mathbb{B}(\mathscr{H})$ of bounded linear operators acting on a Hilbert space $\mathscr{H}$. Among other things, we investigate the relationship between approximate parallelism and norm of inner derivations on $\mathbb{B}(\mathscr{H})$. We also characterize the parallel elements of a $C^*$-algebra by using states. Finally we utilize the linking algebra to give some equivalence assertions regarding parallel elements in a Hilbert $C^*$-module. Keywords:$C^*$-algebra, approximate parallelism, operator parallelism, Hilbert $C^*$-moduleCategories:47A30, 46L05, 46L08, 47B47, 15A60

23. CMB 2014 (vol 58 pp. 128)

Marković, Marijan
 A Sharp Constant for the Bergman Projection For the Bergman projection operator $P$ we prove that \begin{equation*} \|P\colon L^1(B,d\lambda)\rightarrow B_1\| = \frac {(2n+1)!}{n!}. \end{equation*} Here $\lambda$ stands for the hyperbolic metric in the unit ball $B$ of $\mathbb{C}^n$, and $B_1$ denotes the Besov space with an adequate semi--norm. We also consider a generalization of this result. This generalizes some recent results due to PerÃ¤lÃ¤. Keywords:Bergman projections, Besov spacesCategories:45P05, 47B35

24. CMB 2014 (vol 58 pp. 297)

Khamsi, M. A.
 Approximate Fixed Point Sequences of Nonlinear Semigroup in Metric Spaces In this paper, we investigate the common approximate fixed point sequences of nonexpansive semigroups of nonlinear mappings $\{T_t\}_{t \geq 0}$, i.e., a family such that $T_0(x)=x$, $T_{s+t}=T_s(T_t(x))$, where the domain is a metric space $(M,d)$. In particular we prove that under suitable conditions, the common approximate fixed point sequences set is the same as the common approximate fixed point sequences set of two mappings from the family. Then we use the Ishikawa iteration to construct a common approximate fixed point sequence of nonexpansive semigroups of nonlinear mappings. Keywords:approximate fixed point, fixed point, hyperbolic metric space, Ishikawa iterations, nonexpansive mapping, semigroup of mappings, uniformly convex hyperbolic spaceCategories:47H09, 46B20, 47H10, 47E10

25. CMB 2014 (vol 57 pp. 780)

Erzakova, Nina A.
 Measures of Noncompactness in Regular Spaces Previous results by the author on the connection between three of measures of non-compactness obtained for $L_p$, are extended to regular spaces of measurable functions. An example of advantage in some cases one of them in comparison with another is given. Geometric characteristics of regular spaces are determined. New theorems for $(k,\beta)$-boundedness of partially additive operators are proved. Keywords:measure of non-compactness, condensing map, partially additive operator, regular space, ideal spaceCategories:47H08, 46E30, 47H99, 47G10
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