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1. CJM 2014 (vol 66 pp. 1110)

Li, Dong; Xu, Guixiang; Zhang, Xiaoyi
 On the Dispersive Estimate for the Dirichlet SchrÃ¶dinger Propagator and Applications to Energy Critical NLS We consider the obstacle problem for the SchrÃ¶dinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under the radial symmetry we compute explicitly the fundamental solution for the linear Dirichlet SchrÃ¶dinger propagator $e^{it\Delta_D}$ and give a robust algorithm to prove sharp $L^1 \rightarrow L^{\infty}$ dispersive estimates. We showcase the analysis in dimensions $n=5,7$. As an application, we obtain global well-posedness and scattering for defocusing energy-critical NLS on $\Omega=\mathbb{R}^n\backslash \overline{B(0,1)}$ with Dirichlet boundary condition and radial data in these dimensions. Keywords:Dirichlet SchrÃ¶dinger propagator, dispersive estimate, Dirichlet boundary condition, scattering theory, energy criticalCategories:35P25, 35Q55, 47J35

2. CJM 2013 (vol 67 pp. 132)

Clouâtre, Raphaël
 Unitary Equivalence and Similarity to Jordan Models for Weak Contractions of Class $C_0$ We obtain results on the unitary equivalence of weak contractions of class $C_0$ to their Jordan models under an assumption on their commutants. In particular, our work addresses the case of arbitrary finite multiplicity. The main tool is the theory of boundary representations due to Arveson. We also generalize and improve previously known results concerning unitary equivalence and similarity to Jordan models when the minimal function is a Blaschke product. Keywords:weak contractions, operators of class $C_0$, Jordan model, unitary equivalenceCategories:47A45, 47L55

3. CJM 2013 (vol 66 pp. 1143)

Plevnik, Lucijan; Šemrl, Peter
 Maps Preserving Complementarity of Closed Subspaces of a Hilbert Space Let $\mathcal{H}$ and $\mathcal{K}$ be infinite-dimensional separable Hilbert spaces and ${\rm Lat}\,\mathcal{H}$ the lattice of all closed subspaces oh $\mathcal{H}$. We describe the general form of pairs of bijective maps $\phi , \psi : {\rm Lat}\,\mathcal{H} \to {\rm Lat}\,\mathcal{K}$ having the property that for every pair $U,V \in {\rm Lat}\,\mathcal{H}$ we have $\mathcal{H} = U \oplus V \iff \mathcal{K} = \phi (U) \oplus \psi (V)$. Then we reformulate this theorem as a description of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known structural results for maps on idempotents are easy consequences. Keywords:Hilbert space, lattice of closed subspaces, complemented subspaces, adjacent subspaces, idempotentsCategories:46B20, 47B49

4. CJM 2013 (vol 65 pp. 1005)

Forrest, Brian; Miao, Tianxuan
 Uniformly Continuous Functionals and M-Weakly Amenable Groups Let $G$ be a locally compact group. Let $A_{M}(G)$ ($A_{0}(G)$)denote the closure of $A(G)$, the Fourier algebra of $G$ in the space of bounded (completely bounded) multipliers of $A(G)$. We call a locally compact group M-weakly amenable if $A_M(G)$ has a bounded approximate identity. We will show that when $G$ is M-weakly amenable, the algebras $A_{M}(G)$ and $A_{0}(G)$ have properties that are characteristic of the Fourier algebra of an amenable group. Along the way we show that the sets of tolopolically invariant means associated with these algebras have the same cardinality as those of the Fourier algebra. Keywords:Fourier algebra, multipliers, weakly amenable, uniformly continuous functionalsCategories:43A07, 43A22, 46J10, 47L25

5. CJM 2013 (vol 66 pp. 387)

Mashreghi, J.; Shabankhah, M.
 Composition of Inner Functions We study the image of the model subspace $K_\theta$ under the composition operator $C_\varphi$, where $\varphi$ and $\theta$ are inner functions, and find the smallest model subspace which contains the linear manifold $C_\varphi K_\theta$. Then we characterize the case when $C_\varphi$ maps $K_\theta$ into itself. This case leads to the study of the inner functions $\varphi$ and $\psi$ such that the composition $\psi\circ\varphi$ is a divisor of $\psi$ in the family of inner functions. Keywords:composition operators, inner functions, Blaschke products, model subspacesCategories:30D55, 30D05, 47B33

6. CJM 2013 (vol 65 pp. 783)

Garcés, Jorge J.; Peralta, Antonio M.
 Generalised Triple Homomorphisms and Derivations We introduce generalised triple homomorphism between Jordan Banach triple systems as a concept which extends the notion of generalised homomorphism between Banach algebras given by K. Jarosz and B.E. Johnson in 1985 and 1987, respectively. We prove that every generalised triple homomorphism between JB$^*$-triples is automatically continuous. When particularised to C$^*$-algebras, we rediscover one of the main theorems established by B.E. Johnson. We shall also consider generalised triple derivations from a Jordan Banach triple $E$ into a Jordan Banach triple $E$-module, proving that every generalised triple derivation from a JB$^*$-triple $E$ into itself or into $E^*$ is automatically continuous. Keywords:generalised homomorphism, generalised triple homomorphism, generalised triple derivation, Banach algebra, Jordan Banach triple, C$^*$-algebra, JB$^*$-tripleCategories:46L05, 46L70, 47B48, 17C65, 46K70, 46L40, 47B47, 47B49

7. CJM 2013 (vol 66 pp. 641)

Grigor'yan, Alexander; Hu, Jiaxin
 Heat Kernels and Green Functions on Metric Measure Spaces We prove that, in a setting of local Dirichlet forms on metric measure spaces, a two-sided sub-Gaussian estimate of the heat kernel is equivalent to the conjunction of the volume doubling propety, the elliptic Harnack inequality and a certain estimate of the capacity between concentric balls. The main technical tool is the equivalence between the capacity estimate and the estimate of a mean exit time in a ball, that uses two-sided estimates of a Green function in a ball. Keywords:Dirichlet form, heat kernel, Green function, capacityCategories:35K08, 28A80, 31B05, 35J08, 46E35, 47D07

8. CJM 2012 (vol 65 pp. 768)

 Nonself-adjoint Semicrossed Products by Abelian Semigroups Let $\mathcal{S}$ be the semigroup $\mathcal{S}=\sum^{\oplus k}_{i=1}\mathcal{S}_i$, where for each $i\in I$, $\mathcal{S}_i$ is a countable subsemigroup of the additive semigroup $\mathbb{R}_+$ containing $0$. We consider representations of $\mathcal{S}$ as contractions $\{T_s\}_{s\in\mathcal{S}}$ on a Hilbert space with the Nica-covariance property: $T_s^*T_t=T_tT_s^*$ whenever $t\wedge s=0$. We show that all such representations have a unique minimal isometric Nica-covariant dilation. This result is used to help analyse the nonself-adjoint semicrossed product algebras formed from Nica-covariant representations of the action of $\mathcal{S}$ on an operator algebra $\mathcal{A}$ by completely contractive endomorphisms. We conclude by calculating the $C^*$-envelope of the isometric nonself-adjoint semicrossed product algebra (in the sense of Kakariadis and Katsoulis). Keywords:semicrossed product, crossed product, C*-envelope, dilationsCategories:47L55, 47A20, 47L65

9. CJM 2011 (vol 64 pp. 1329)

Izuchi, Kei Ji; Nguyen, Quang Dieu; Ohno, Shûichi
 Composition Operators Induced by Analytic Maps to the Polydisk We study properties of composition operators induced by symbols acting from the unit disk to the polydisk. This result will be involved in the investigation of weighted composition operators on the Hardy space on the unit disk and moreover be concerned with composition operators acting from the Bergman space to the Hardy space on the unit disk. Keywords:composition operators, Hardy spaces, polydiskCategories:47B33, 32A35, 30H10

10. CJM 2011 (vol 63 pp. 1161)

Neuwirth, Stefan; Ricard, Éric
 Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue-Orlicz spaces of a discrete group $\varGamma$ and relative Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum $\varLambda\subseteq\varGamma$, the norm of the Hilbert transform and the Riesz projection on Schatten-von-Neumann classes with exponent a power of 2, and the norm of Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less than 1. Keywords:Fourier multiplier, Toeplitz Schur multiplier, lacunary set, unconditional approximation property, Hilbert transform, Riesz projectionCategories:47B49, 43A22, 43A46, 46B28

11. CJM 2011 (vol 64 pp. 183)

 Negative Powers of Laguerre Operators We study negative powers of Laguerre differential operators in $\mathbb{R}^d$, $d\ge1$. For these operators we prove two-weight $L^p-L^q$ estimates with ranges of $q$ depending on $p$. The case of the harmonic oscillator (Hermite operator) has recently been treated by Bongioanni and Torrea by using a straightforward approach of kernel estimates. Here these results are applied in certain Laguerre settings. The procedure is fairly direct for Laguerre function expansions of Hermite type, due to some monotonicity properties of the kernels involved. The case of Laguerre function expansions of convolution type is less straightforward. For half-integer type indices $\alpha$ we transfer the desired results from the Hermite setting and then apply an interpolation argument based on a device we call the convexity principle to cover the continuous range of $\alpha\in[-1/2,\infty)^d$. Finally, we investigate negative powers of the Dunkl harmonic oscillator in the context of a finite reflection group acting on $\mathbb{R}^d$ and isomorphic to $\mathbb Z^d_2$. The two weight $L^p-L^q$ estimates we obtain in this setting are essentially consequences of those for Laguerre function expansions of convolution type. Keywords:potential operator, fractional integral, Riesz potential, negative power, harmonic oscillator, Laguerre operator, Dunkl harmonic oscillatorCategories:47G40, 31C15, 26A33

12. CJM 2011 (vol 63 pp. 1188)

Śliwa, Wiesław; Ziemkowska, Agnieszka
 On Complemented Subspaces of Non-Archimedean Power Series Spaces The non-archimedean power series spaces, $A_1(a)$ and $A_\infty(b)$, are the best known and most important examples of non-archimedean nuclear FrÃ©chet spaces. We prove that the range of every continuous linear map from $A_p(a)$ to $A_q(b)$ has a Schauder basis if either $p=1$ or $p=\infty$ and the set $M_{b,a}$ of all bounded limit points of the double sequence $(b_i/a_j)_{i,j\in\mathbb{N}}$ is bounded. It follows that every complemented subspace of a power series space $A_p(a)$ has a Schauder basis if either $p=1$ or $p=\infty$ and the set $M_{a,a}$ is bounded. Keywords:non-archimedean KÃ¶the space, range of a continuous linear map, Schauder basisCategories:46S10, 47S10, 46A35

13. CJM 2011 (vol 63 pp. 862)

Hosokawa, Takuya; Nieminen, Pekka J.; Ohno, Shûichi
 Linear Combinations of Composition Operators on the Bloch Spaces We characterize the compactness of linear combinations of analytic composition operators on the Bloch space. We also study their boundedness and compactness on the little Bloch space. Keywords: composition operator, compactness, Bloch spaceCategories:47B33, 30D45, 47B07

14. CJM 2011 (vol 63 pp. 648)

Ngai, Sze-Man
 Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps We set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition. Keywords:spectral dimension, fractal, Laplacian, self-similar measure, iterated function system with overlaps, second-order self-similar identitiesCategories:28A80, , , , 35P20, 35J05, 43A05, 47A75

15. CJM 2011 (vol 63 pp. 533)

Espínola, Rafa; Fernández-León, Aurora
 On Best Proximity Points in Metric and Banach Spaces In this paper we study the existence and uniqueness of best proximity points of cyclic contractions as well as the convergence of iterates to such proximity points. We take two different approaches, each one leading to different results that complete, if not improve, other similar results in the theory. Results in this paper stand for Banach spaces, geodesic metric spaces and metric spaces. We also include an appendix on CAT$(0)$ spaces where we study the particular behavior of these spaces regarding the problems we are concerned with. Categories:54H25, 47H09

16. CJM 2010 (vol 62 pp. 1419)

Yang, Dachun; Yang, Dongyong
 BMO-Estimates for Maximal Operators via Approximations of the Identity with Non-Doubling Measures Let $\mu$ be a nonnegative Radon measure on $\mathbb{R}^d$ that satisfies the growth condition that there exist constants $C_0>0$ and $n\in(0,d]$ such that for all $x\in\mathbb{R}^d$ and $r>0$, ${\mu(B(x,\,r))\le C_0r^n}$, where $B(x,r)$ is the open ball centered at $x$ and having radius $r$. In this paper, the authors prove that if $f$ belongs to the $\textrm {BMO}$-type space $\textrm{RBMO}(\mu)$ of Tolsa, then the homogeneous maximal function $\dot{\mathcal{M}}_S(f)$ (when $\mathbb{R}^d$ is not an initial cube) and the inhomogeneous maximal function $\mathcal{M}_S(f)$ (when $\mathbb{R}^d$ is an initial cube) associated with a given approximation of the identity $S$ of Tolsa are either infinite everywhere or finite almost everywhere, and in the latter case, $\dot{\mathcal{M}}_S$ and $\mathcal{M}_S$ are bounded from $\textrm{RBMO}(\mu)$ to the $\textrm {BLO}$-type space $\textrm{RBLO}(\mu)$. The authors also prove that the inhomogeneous maximal operator $\mathcal{M}_S$ is bounded from the local $\textrm {BMO}$-type space $\textrm{rbmo}(\mu)$ to the local $\textrm {BLO}$-type space $\textrm{rblo}(\mu)$. Keywords:Non-doubling measure, maximal operator, approximation of the identity, RBMO(mu), RBLO(mu), rbmo(mu), rblo(mu)Categories:42B25, 42B30, 47A30, 43A99

17. CJM 2010 (vol 62 pp. 889)

Xia, Jingbo
 Singular Integral Operators and Essential Commutativity on the Sphere Let ${\mathcal T}$ be the $C^\ast$-algebra generated by the Toeplitz operators $\{T_\varphi : \varphi \in L^\infty (S,d\sigma )\}$ on the Hardy space $H^2(S)$ of the unit sphere in $\mathbf{C}^n$. It is well known that ${\mathcal T}$ is contained in the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$. We show that the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$ is strictly larger than ${\mathcal T}$. Categories:32A55, 46L05, 47L80

18. CJM 2009 (vol 62 pp. 74)

Ducrot, Arnaud; Liu, Zhihua; Magal, Pierre
 Projectors on the Generalized Eigenspaces for Neutral Functional Differential Equations in $L^{p}$ Spaces We present the explicit formulas for the projectors on the generalized eigenspaces associated with some eigenvalues for linear neutral functional differential equations (NFDE) in $L^{p}$ spaces by using integrated semigroup theory. The analysis is based on the main result established elsewhere by the authors and results by Magal and Ruan on non-densely defined Cauchy problem. We formulate the NFDE as a non-densely defined Cauchy problem and obtain some spectral properties from which we then derive explicit formulas for the projectors on the generalized eigenspaces associated with some eigenvalues. Such explicit formulas are important in studying bifurcations in some semi-linear problems. Keywords:neutral functional differential equations, semi-linear problem, integrated semigroup, spectrum, projectorsCategories:34K05, 35K57, 47A56, 47H20

19. CJM 2009 (vol 62 pp. 415)

Sun, Shunhua; Zheng, Dechao; Zhong, Changyong
 Classification of Reducing Subspaces of a Class of Multiplication Operators on the Bergman Space via the Hardy Space of the Bidisk In this paper we obtain a complete description of nontrivial minimal reducing subspaces of the multiplication operator by a Blaschke product with four zeros on the Bergman space of the unit disk via the Hardy space of the bidisk. Categories:47B35, 47B38

20. CJM 2009 (vol 62 pp. 439)

Sundhäll, Marcus; Tchoundja, Edgar
 On Hankel Forms of Higher Weights: The Case of Hardy Spaces In this paper we study bilinear Hankel forms of higher weights on Hardy spaces in several dimensions. (The Schatten class Hankel forms of higher weights on weighted Bergman spaces have already been studied by Janson and Peetre for one dimension and by SundhÃ¤ll for several dimensions). We get a full characterization of Schatten class Hankel forms in terms of conditions for the symbols to be in certain Besov spaces. Also, the Hankel forms are bounded and compact if and only if the symbols satisfy certain Carleson measure criteria and vanishing Carleson measure criteria, respectively. Keywords:Hankel forms, Schattenâvon Neumann classes, Bergman spaces, Hardy spaces, Besov spaces, transvectant, unitary representations, MÃ¶bius groupCategories:32A25, 32A35, 32A37, 47B35

21. CJM 2009 (vol 62 pp. 305)

Hua, He; Yunbai, Dong; Xianzhou, Guo
 Approximation and Similarity Classification of Stably Finitely Strongly Irreducible Decomposable Operators Let $\mathcal H$ be a complex separable Hilbert space and ${\mathcal L}({\mathcal H})$ denote the collection of bounded linear operators on ${\mathcal H}$. In this paper, we show that for any operator $A\in{\mathcal L}({\mathcal H})$, there exists a stably finitely (SI) decomposable operator $A_\epsilon$, such that $\|A-A_{\epsilon}\|<\epsilon$ and ${\mathcal{\mathcal A}'(A_{\epsilon})}/\operatorname{rad} {{\mathcal A}'(A_{\epsilon})}$ is commutative, where $\operatorname{rad}{{\mathcal A}'(A_{\epsilon})}$ is the Jacobson radical of ${{\mathcal A}'(A_{\epsilon})}$. Moreover, we give a similarity classification of the stably finitely decomposable operators that generalizes the result on similarity classification of Cowen-Douglas operators given by C. L. Jiang. Keywords:$K_{0}$-group, strongly irreducible decomposition, CowenâDouglas operators, commutant algebra, similarity classificationCategories:47A05, 47A55, 46H20

22. CJM 2009 (vol 62 pp. 242)

Azagra, Daniel; Fry, Robb
 A Second Order Smooth Variational Principle on Riemannian Manifolds We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature. Keywords:smooth variational principle, Riemannian manifoldCategories:58E30, 49J52, 46T05, 47J30, 58B20

23. CJM 2009 (vol 62 pp. 133)

Makarov, Konstantin A.; Skripka, Anna
 Some Applications of the Perturbation Determinant in Finite von Neumann Algebras In the finite von Neumann algebra setting, we introduce the concept of a perturbation determinant associated with a pair of self-adjoint elements $H_0$ and $H$ in the algebra and relate it to the concept of the de la Harpe--Skandalis homotopy invariant determinant associated with piecewise $C^1$-paths of operators joining $H_0$ and $H$. We obtain an analog of Krein's formula that relates the perturbation determinant and the spectral shift function and, based on this relation, we derive subsequently (i) the Birman--Solomyak formula for a general non-linear perturbation, (ii) a universality of a spectral averaging, and (iii) a generalization of the Dixmier--Fuglede--Kadison differentiation formula. Keywords:perturbation determinant, trace formulae, von Neumann algebrasCategories:47A55, 47C15, 47A53

24. CJM 2009 (vol 61 pp. 1239)

Davidson, Kenneth R.; Yang, Dilian
 Periodicity in Rank 2 Graph Algebras Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of $\mathrm{C}^*(\mathbb{F}^+_{\theta})$. The periodic $\mathrm{C}^*$-algebras are characterized, and it is shown that $\mathrm{C}^*(\mathbb{F}^+_{\theta}) \simeq \mathrm{C}(\mathbb{T})\otimes\mathfrak{A}$ where $\mathfrak{A}$ is a simple $\mathrm{C}^*$-algebra. Keywords:higher rank graph, aperiodicity condition, simple $\mathrm{C}^*$-algebra, expectationCategories:47L55, 47L30, 47L75, 46L05

25. CJM 2009 (vol 61 pp. 282)

Bouya, Brahim
 Closed Ideals in Some Algebras of Analytic Functions We obtain a complete description of closed ideals of the algebra $\cD\cap \cL$, $0<\alpha\leq\frac{1}{2}$, where $\cD$ is the Dirichlet space and $\cL$ is the algebra of analytic functions satisfying the Lipschitz condition of order $\alpha$. Categories:46E20, 30H05, 47A15
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