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1. CMB Online first

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

2. CMB Online first

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

3. CMB Online first

Fang, Zhong-Shan; Zhou, Ze-Hua
 New Characterizations of the Weighted Composition Operators Between Bloch Type Spaces in the Polydisk We give some new characterizations for compactness of weighted composition operators $uC_\varphi$ acting on Bloch-type spaces in terms of the power of the components of $\varphi,$ where $\varphi$ is a holomorphic self-map of the polydisk $\mathbb{D}^n,$ thus generalizing the results obtained by HyvÃ¤rinen and LindstrÃ¶m in 2012. Keywords:weighted composition operator, compactness, Bloch type spaces, polydisk, several complex variablesCategories:47B38, 47B33, 32A37, 45P05, 47G10

4. CMB Online first

Bownik, Marcin; Jasper, John
 Constructive Proof of Carpenter's Theorem We give a constructive proof of Carpenter's Theorem due to Kadison. Unlike the original proof our approach also yields the real case of this theorem. Keywords:diagonals of projections, the Schur-Horn theorem, the Pythagorean theorem, the Carpenter theorem, spectral theoryCategories:42C15, 47B15, 46C05

5. CMB 2013 (vol 57 pp. 270)

Didas, Michael; Eschmeier, Jörg
 Derivations on Toeplitz Algebras Let $H^2(\Omega)$ be the Hardy space on a strictly pseudoconvex domain $\Omega \subset \mathbb{C}^n$, and let $A \subset L^\infty(\partial \Omega)$ denote the subalgebra of all $L^\infty$-functions $f$ with compact Hankel operator $H_f$. Given any closed subalgebra $B \subset A$ containing $C(\partial \Omega)$, we describe the first Hochschild cohomology group of the corresponding Toeplitz algebra $\mathcal(B) \subset B(H^2(\Omega))$. In particular, we show that every derivation on $\mathcal{T}(A)$ is inner. These results are new even for $n=1$, where it follows that every derivation on $\mathcal{T}(H^\infty+C)$ is inner, while there are non-inner derivations on $\mathcal{T}(H^\infty+C(\partial \mathbb{B}_n))$ over the unit ball $\mathbb{B}_n$ in dimension $n\gt 1$. Keywords:derivations, Toeplitz algebras, strictly pseudoconvex domainsCategories:47B47, 47B35, 47L80

6. CMB 2012 (vol 57 pp. 166)

Öztop, Serap; Spronk, Nico
 On Minimal and Maximal $p$-operator Space Structures We show that for $p$-operator spaces, there are natural notions of minimal and maximal structures. These are useful for dealing with tensor products. Keywords:$p$-operator space, min space, max spaceCategories:46L07, 47L25, 46G10

7. CMB 2012 (vol 57 pp. 80)

Khemphet, Anchalee; Peters, Justin R.
 Semicrossed Products of the Disk Algebra and the Jacobson Radical We consider semicrossed products of the disk algebra with respect to endomorphisms defined by finite Blaschke products. We characterize the Jacobson radical of these operator algebras. Furthermore, in the case the finite Blaschke product is elliptic, we show that the semicrossed product contains no nonzero quasinilpotent elements. However, if the finite Blaschke product is hyperbolic or parabolic with positive hyperbolic step, the Jacobson radical is nonzero and a proper subset of the set of quasinilpotent elements. Keywords:semicrossed product, disk algebra, Jacobson radicalCategories:47L65, 47L20, 30J10, 30H50

8. CMB 2012 (vol 56 pp. 477)

Ayadi, Adlene
 Hypercyclic Abelian Groups of Affine Maps on $\mathbb{C}^{n}$ We give a characterization of hypercyclic abelian group $\mathcal{G}$ of affine maps on $\mathbb{C}^{n}$. If $\mathcal{G}$ is finitely generated, this characterization is explicit. We prove in particular that no abelian group generated by $n$ affine maps on $\mathbb{C}^{n}$ has a dense orbit. Keywords:affine, hypercyclic, dense, orbit, affine group, abelianCategories:37C85, 47A16

9. CMB 2012 (vol 57 pp. 145)

Mustafayev, H. S.
 The Essential Spectrum of the Essentially Isometric Operator Let $T$ be a contraction on a complex, separable, infinite dimensional Hilbert space and let $\sigma \left( T\right)$ (resp. $\sigma _{e}\left( T\right) )$ be its spectrum (resp. essential spectrum). We assume that $T$ is an essentially isometric operator, that is $I_{H}-T^{\ast }T$ is compact. We show that if $D\diagdown \sigma \left( T\right) \neq \emptyset ,$ then for every $f$ from the disc-algebra, \begin{equation*} \sigma _{e}\left( f\left( T\right) \right) =f\left( \sigma _{e}\left( T\right) \right) , \end{equation*} where $D$ is the open unit disc. In addition, if $T$ lies in the class $C_{0\cdot }\cup C_{\cdot 0},$ then \begin{equation*} \sigma _{e}\left( f\left( T\right) \right) =f\left( \sigma \left( T\right) \cap \Gamma \right) , \end{equation*} where $\Gamma$ is the unit circle. Some related problems are also discussed. Keywords:Hilbert space, contraction, essentially isometric operator, (essential) spectrum, functional calculusCategories:47A10, 47A53, 47A60, 47B07

10. CMB 2012 (vol 57 pp. 25)

Bourin, Jean-Christophe; Harada, Tetsuo; Lee, Eun-Young
 Subadditivity Inequalities for Compact Operators Some subadditivity inequalities for matrices and concave functions also hold for Hilbert space operators, but (unfortunately!) with an additional $\varepsilon$ term. It seems not possible to erase this residual term. However, in case of compact operators we show that the $\varepsilon$ term is unnecessary. Further, these inequalities are strict in a certain sense when some natural assumptions are satisfied. The discussion also stresses on matrices and their compressions and several open questions or conjectures are considered, both in the matrix and operator settings. Keywords:concave or convex function, Hilbert space, unitary orbits, compact operators, compressions, matrix inequalitiesCategories:47A63, 15A45

11. CMB 2011 (vol 56 pp. 459)

Athavale, Ameer; Patil, Pramod
 On Certain Multivariable Subnormal Weighted Shifts and their Duals To every subnormal $m$-variable weighted shift $S$ (with bounded positive weights) corresponds a positive Reinhardt measure $\mu$ supported on a compact Reinhardt subset of $\mathbb C^m$. We show that, for $m \geq 2$, the dimensions of the $1$-st cohomology vector spaces associated with the Koszul complexes of $S$ and its dual ${\tilde S}$ are different if a certain radial function happens to be integrable with respect to $\mu$ (which is indeed the case with many classical examples). In particular, $S$ cannot in that case be similar to ${\tilde S}$. We next prove that, for $m \geq 2$, a Fredholm subnormal $m$-variable weighted shift $S$ cannot be similar to its dual. Keywords:subnormal, Reinhardt, Betti numbersCategory:47B20

12. CMB 2011 (vol 56 pp. 593)

Liu, Congwen; Zhou, Lifang
 On the $p$-norm of an Integral Operator in the Half Plane We give a partial answer to a conjecture of DostaniÄ on the determination of the norm of a class of integral operators induced by the weighted Bergman projection in the upper half plane. Keywords:Bergman projection, integral operator, $L^p$-norm, the upper half planeCategories:47B38, 47G10, 32A36

13. CMB 2011 (vol 56 pp. 39)

Ben Amara, Jamel
 Comparison Theorem for Conjugate Points of a Fourth-order Linear Differential Equation In 1961, J. Barrett showed that if the first conjugate point $\eta_1(a)$ exists for the differential equation $(r(x)y'')''= p(x)y,$ where $r(x)\gt 0$ and $p(x)\gt 0$, then so does the first systems-conjugate point $\widehat\eta_1(a)$. The aim of this note is to extend this result to the general equation with middle term $(q(x)y')'$ without further restriction on $q(x)$, other than continuity. Keywords:fourth-order linear differential equation, conjugate points, system-conjugate points, subwronskiansCategories:47E05, 34B05, 34C10

14. CMB 2011 (vol 56 pp. 400)

Prunaru, Bebe
 A Factorization Theorem for Multiplier Algebras of Reproducing Kernel Hilbert Spaces Let $(X,\mathcal B,\mu)$ be a $\sigma$-finite measure space and let $H\subset L^2(X,\mu)$ be a separable reproducing kernel Hilbert space on $X$. We show that the multiplier algebra of $H$ has property $(A_1(1))$. Keywords:reproducing kernel Hilbert space, Berezin transform, dual algebraCategories:46E22, 47B32, 47L45

15. CMB 2011 (vol 56 pp. 229)

Arvanitidis, Athanasios G.; Siskakis, Aristomenis G.
 CesÃ ro Operators on the Hardy Spaces of the Half-Plane In this article we study the CesÃ ro operator $$\mathcal{C}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\zeta)\,d\zeta,$$ and its companion operator $\mathcal{T}$ on Hardy spaces of the upper half plane. We identify $\mathcal{C}$ and $\mathcal{T}$ as resolvents for appropriate semigroups of composition operators and we find the norm and the spectrum in each case. The relation of $\mathcal{C}$ and $\mathcal{T}$ with the corresponding Ces\{a}ro operators on Lebesgue spaces $L^p(\mathbb R)$ of the boundary line is also discussed. Keywords:CesÃ ro operators, Hardy spaces, semigroups, composition operatorsCategories:47B38, 30H10, 47D03

16. CMB 2011 (vol 55 pp. 646)

Zhou, Jiang; Ma, Bolin
 Marcinkiewicz Commutators with Lipschitz Functions in Non-homogeneous Spaces Under the assumption that $\mu$ is a nondoubling measure, we study certain commutators generated by the Lipschitz function and the Marcinkiewicz integral whose kernel satisfies a HÃ¶rmander-type condition. We establish the boundedness of these commutators on the Lebesgue spaces, Lipschitz spaces, and Hardy spaces. Our results are extensions of known theorems in the doubling case. Keywords:non doubling measure, Marcinkiewicz integral, commutator, ${\rm Lip}_{\beta}(\mu)$, $H^1(\mu)$Categories:42B25, 47B47, 42B20, 47A30

17. CMB 2011 (vol 55 pp. 673)

Aizenbud, Avraham; Gourevitch, Dmitry
 Multiplicity Free Jacquet Modules Let $F$ be a non-Archimedean local field or a finite field. Let $n$ be a natural number and $k$ be $1$ or $2$. Consider $G:=\operatorname{GL}_{n+k}(F)$ and let $M:=\operatorname{GL}_n(F) \times \operatorname{GL}_k(F)\lt G$ be a maximal Levi subgroup. Let $U\lt G$ be the corresponding unipotent subgroup and let $P=MU$ be the corresponding parabolic subgroup. Let $J:=J_M^G: \mathcal{M}(G) \to \mathcal{M}(M)$ be the Jacquet functor, i.e., the functor of coinvariants with respect to $U$. In this paper we prove that $J$ is a multiplicity free functor, i.e., $\dim \operatorname{Hom}_M(J(\pi),\rho)\leq 1$, for any irreducible representations $\pi$ of $G$ and $\rho$ of $M$. We adapt the classical method of Gelfand and Kazhdan, which proves the multiplicity free" property of certain representations to prove the `multiplicity free" property of certain functors. At the end we discuss whether other Jacquet functors are multiplicity free. Keywords:multiplicity one, Gelfand pair, invariant distribution, finite groupCategories:20G05, 20C30, 20C33, 46F10, 47A67

18. CMB 2011 (vol 55 pp. 555)

Michalowski, Nicholas; Rule, David J.; Staubach, Wolfgang
 Weighted $L^p$ Boundedness of Pseudodifferential Operators and Applications In this paper we prove weighted norm inequalities with weights in the $A_p$ classes, for pseudodifferential operators with symbols in the class ${S^{n(\rho -1)}_{\rho, \delta}}$ that fall outside the scope of CalderÃ³n-Zygmund theory. This is accomplished by controlling the sharp function of the pseudodifferential operator by Hardy-Littlewood type maximal functions. Our weighted norm inequalities also yield $L^{p}$ boundedness of commutators of functions of bounded mean oscillation with a wide class of operators in $\mathrm{OP}S^{m}_{\rho, \delta}$. Keywords:weighted norm inequality, pseudodifferential operator, commutator estimatesCategories:42B20, 42B25, 35S05, 47G30

19. CMB 2011 (vol 54 pp. 654)

Forrest, Brian E.; Runde, Volker
 Norm One Idempotent $cb$-Multipliers with Applications to the Fourier Algebra in the $cb$-Multiplier Norm For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely bounded multipliers of $A(G)$, and let $A_{\mathit{Mcb}}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \subset G$ is a norm one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we describe the closed ideals of $A_{\mathit{Mcb}}(G)$ with an approximate identity bounded by $1$, and we characterize those $G$ for which $A_{\mathit{Mcb}}(G)$ is $1$-amenable in the sense of B. E. Johnson. (We can even slightly relax the norm bounds.) Keywords:amenability, bounded approximate identity, $cb$-multiplier norm, Fourier algebra, norm one idempotentCategories:43A22, 20E05, 43A30, 46J10, 46J40, 46L07, 47L25

20. CMB 2011 (vol 54 pp. 456)

Gustafson, Karl
 On Operator Sum and Product Adjoints and Closures We comment on domain conditions that regulate when the adjoint of the sum or product of two unbounded operators is the sum or product of their adjoints, and related closure issues. The quantum mechanical problem PHP essentially selfadjoint for unbounded Hamiltonians is addressed, with new results. Keywords:unbounded operators, adjoints of sums and products, quantum mechanicsCategory:47A05

21. CMB 2011 (vol 55 pp. 15)

Akiyama, Shigeki; Suzuki, Tomonari
 Browder's Convergence for One-Parameter Nonexpansive Semigroups We give the sufficient and necessary conditions of Browder's convergence theorem for one-parameter nonexpansive semigroups which was proved by Suzuki. We also discuss the perfect kernels of topological spaces. Keywords:nonexpansive semigroup, common fixed point, Browder's convergence, perfect kernelCategory:47H20

22. CMB 2011 (vol 55 pp. 882)

Xueli, Song; Jigen, Peng
 Equivalence of $L_p$ Stability and Exponential Stability of Nonlinear Lipschitzian Semigroups $L_p$ stability and exponential stability are two important concepts for nonlinear dynamic systems. In this paper, we prove that a nonlinear exponentially bounded Lipschitzian semigroup is exponentially stable if and only if the semigroup is $L_p$ stable for some $p>0$. Based on the equivalence, we derive two sufficient conditions for exponential stability of the nonlinear semigroup. The results obtained extend and improve some existing ones. Keywords:exponentially stable, $L_p$ stable, nonlinear Lipschitzian semigroupsCategories:34D05, 47H20

23. CMB 2011 (vol 54 pp. 506)

Neamaty, A.; Mosazadeh, S.
 On the Canonical Solution of the Sturm-Liouville Problem with Singularity and Turning Point of Even Order In this paper, we are going to investigate the canonical property of solutions of systems of differential equations having a singularity and turning point of even order. First, by a replacement, we transform the system to the Sturm-Liouville equation with turning point. Using of the asymptotic estimates provided by Eberhard, Freiling, and Schneider for a special fundamental system of solutions of the Sturm-Liouville equation, we study the infinite product representation of solutions of the systems. Then we transform the Sturm-Liouville equation with turning point to the equation with singularity, then we study the asymptotic behavior of its solutions. Such representations are relevant to the inverse spectral problem. Keywords:turning point, singularity, Sturm-Liouville, infinite products, Hadamard's theorem, eigenvaluesCategories:34B05, 34Lxx, 47E05

24. CMB 2011 (vol 55 pp. 339)

Loring, Terry A.
 From Matrix to Operator Inequalities We generalize LÃ¶wner's method for proving that matrix monotone functions are operator monotone. The relation $x\leq y$ on bounded operators is our model for a definition of $C^{*}$-relations being residually finite dimensional. Our main result is a meta-theorem about theorems involving relations on bounded operators. If we can show there are residually finite dimensional relations involved and verify a technical condition, then such a theorem will follow from its restriction to matrices. Applications are shown regarding norms of exponentials, the norms of commutators, and "positive" noncommutative $*$-polynomials. Keywords:$C*$-algebras, matrices, bounded operators, relations, operator norm, order, commutator, exponential, residually finite dimensionalCategories:46L05, 47B99

25. CMB 2011 (vol 55 pp. 441)

Zorboska, Nina
 Univalently Induced, Closed Range, Composition Operators on the Bloch-type Spaces While there is a large variety of univalently induced closed range composition operators on the Bloch space, we show that the only univalently induced, closed range, composition operators on the Bloch-type spaces $B^{\alpha}$ with $\alpha \ne 1$ are the ones induced by a disc automorphism. Keywords:composition operators, Bloch-type spaces, closed range, univalentCategories:47B35, 32A18
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