1. CMB Online first
2. CMB 2015 (vol 58 pp. 723)
 Castro, Alfonso; Fischer, Emily

Infinitely Many Rotationally Symmetric Solutions to a Class of Semilinear LaplaceBeltrami Equations on Spheres
We show that a class of semilinear LaplaceBeltrami 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
Pohozaevtype
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 $s^{p1}s$ for
$s$ large
with $1 \lt p \lt (n+5)/(n3)$.
Keywords:LaplaceBeltrami operator, semilinear equation, rotational solution, superlinear nonlinearity, subsuper critical nonlinearity Categories:58J05, 35A24 

3. CMB Online first
 Li, ChiKwong; Tsai, MingCheng

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
$\P\ \le \sqrt{a}  \sqrt{b}\sqrt{(1a)(1b)}.$ 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 theorem Categories:47A60, 47A68, 47A63 

4. 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 Opialtype 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, Opialtype inequality, HÃ¶lder's inequality, partial differential operator, partial differential equation Categories:26D10, 35A23 

5. CMB Online first
 Song, Hongxue; Chen, Caisheng; Yan, Qinglun

Existence of multiple solutions for a $p$Laplacian system in $\textbf{R}^{N}$ with signchanging weight functions
In this paper, we consider the quasilinear elliptic
problem
\[
\left\{
\begin{aligned}
&
M
\left(\int_{\mathbb{R}^{N}}x^{ap}\nabla u^{p}dx
\right){\rm
div}
\left(x^{ap}\nabla u^{p2}\nabla u
\right)
\\
&
\qquad=\frac{\alpha}{\alpha+\beta}H(x)u^{\alpha2}uv^{\beta}+\lambda
h_{1}(x)u^{q2}u,
\\
&
M
\left(\int_{\mathbb{R}^{N}}x^{ap}\nabla v^{p}dx
\right){\rm
div}
\left(x^{ap}\nabla v^{p2}\nabla v
\right)
\\
&
\qquad=\frac{\beta}{\alpha+\beta}H(x)v^{\beta2}vu^{\alpha}+\mu
h_{2}(x)v^{q2}v,
\\
&u(x)\gt 0,\quad v(x)\gt 0, \quad x\in \mathbb{R}^{N}
\end{aligned}
\right.
\]
where $\lambda, \mu\gt 0$, $1\lt p\lt N$,
$1\lt q\lt p\lt p(\tau+1)\lt \alpha+\beta\lt p^{*}=\frac{Np}{Np}$, $0\leq
a\lt \frac{Np}{p}$, $a\leq b\lt a+1$, $d=a+1b\gt 0$, $M(s)=k+l s^{\tau}$,
$k\gt 0$, $l, \tau\geq0$ 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 nonlinearities Category:35J66 

6. 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
firstorder Sobolev spaces. Moreover, two key pointwise
inequalities for the partial derivatives of the multisublinear
maximal functions are established. As an application, the
quasicontinuity on the multisublinear maximal function is also
obtained.
Keywords:regularity, multisublinear maximal operator, Sobolev spaces, partial deviative, quasicontinuity Categories:42B25, 46E35 

7. 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 oneparameter groups of surjective isometries Categories:46E15, 47B15, 47B38 

8. CMB 2014 (vol 58 pp. 432)
 Yang, Dachun; Yang, Sibei

Secondorder Riesz Transforms and Maximal Inequalities Associated with Magnetic SchrÃ¶dinger Operators
Let $A:=(\nablai\vec{a})\cdot(\nablai\vec{a})+V$ be a
magnetic SchrÃ¶dinger operator on $\mathbb{R}^n$,
where $\vec{a}:=(a_1,\dots, a_n)\in L^2_{\mathrm{loc}}(\mathbb{R}^n,\mathbb{R}^n)$
and $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ satisfy some reverse
HÃ¶lder conditions.
Let $\varphi\colon \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be such that
$\varphi(x,\cdot)$ for any given $x\in\mathbb{R}^n$ is an Orlicz function,
$\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ for all $t\in (0,\infty)$
(the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index
$I(\varphi)\in(0,1]$. In this article, the authors prove that
secondorder Riesz transforms $VA^{1}$ and
$(\nablai\vec{a})^2A^{1}$ are bounded from the
MusielakOrliczHardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$,
to the MusielakOrlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, the authors
establish the boundedness of $VA^{1}$ on $H_{\varphi, A}(\mathbb{R}^n)$. As applications, some
maximal inequalities associated with $A$ in the scale of $H_{\varphi,
A}(\mathbb{R}^n)$ are obtained.
Keywords:MusielakOrliczHardy space, magnetic SchrÃ¶dinger operator, atom, secondorder Riesz transform, maximal inequality Categories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30 

9. CMB 2014 (vol 58 pp. 51)
 De Nitties, Giuseppe; SchulzBaldes, Hermann

Spectral Flows of Dilations of Fredholm Operators
Given an essentially unitary contraction and an arbitrary unitary
dilation of it, there is a naturally associated spectral flow which is
shown to be equal to the index of the operator. This result is
interpreted in terms of the $K$theory of an associated mapping
cone. It is then extended to connect $\mathbb{Z}_2$ indices of odd symmetric
Fredholm operators to a $\mathbb{Z}_2$valued spectral flow.
Keywords:spectral flow, Fredholm operators, Z2 indices Categories:19K56, 46L80 

10. 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 quasicomplementary, noncomplementary subspaces of
$\ell_\infty$ that are isometric to $\ell_\infty$.
Keywords:$L_1$, Tauberian operator, $\ell_\infty$ Categories:46E30, 46B08, 47A53 

11. CMB 2014 (vol 58 pp. 19)
 Chen, Jiecheng; Hu, Guoen

Compact Commutators of Rough Singular Integral Operators
Let $b\in \mathrm{BMO}(\mathbb{R}^n)$ and $T_{\Omega}$ be the singular
integral operator with kernel $\frac{\Omega(x)}{x^n}$, where
$\Omega$ is homogeneous of degree zero, integrable and has mean
value zero on the unit sphere $S^{n1}$. In this paper, by Fourier
transform estimates and approximation to the operator $T_{\Omega}$
by integral operators with smooth kernels, it is proved that if
$b\in \mathrm{CMO}(\mathbb{R}^n)$ and $\Omega$ satisfies a certain
minimal size condition, then the commutator generated by $b$ and
$T_{\Omega}$ is a compact operator on $L^p(\mathbb{R}^n)$ for
appropriate index $p$. The associated maximal operator is also
considered.
Keywords:commutator,singular integral operator, compact operator, maximal operator Category:42B20 

12. 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 $C^*$module Categories:47A30, 46L05, 46L08, 47B47, 15A60 

13. CMB 2014 (vol 57 pp. 834)
 Koh, Doowon

Restriction Operators Acting on Radial Functions on Vector Spaces Over Finite Fields
We study $L^pL^r$ restriction estimates for
algebraic varieties $V$ in the case when restriction operators act on
radial functions in the finite field setting.
We show that if the varieties $V$ lie in odd dimensional vector
spaces over finite fields, then the conjectured restriction estimates
are possible for all radial test functions.
In addition, assuming that the varieties $V$ are defined in even
dimensional spaces and have few intersection points with the sphere
of zero radius, we also obtain the conjectured exponents for all
radial test functions.
Keywords:finite fields, radial functions, restriction operators Categories:42B05, 43A32, 43A15 

14. 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 noncompactness 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 noncompactness, condensing map, partially additive operator, regular space, ideal space Categories:47H08, 46E30, 47H99, 47G10 

15. CMB 2013 (vol 57 pp. 794)
 Fang, ZhongShan; Zhou, ZeHua

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 Blochtype spaces in
terms of the power of the components of $\varphi,$ where $\varphi$
is a holomorphic selfmap 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 variables Categories:47B38, 47B33, 32A37, 45P05, 47G10 

16. CMB 2013 (vol 57 pp. 821)
 Jeong, Imsoon; Kim, Seonhui; Suh, Young Jin

Real Hypersurfaces in Complex TwoPlane Grassmannians with Reeb Parallel Structure Jacobi Operator
In this paper we give a characterization of a real hypersurface of
Type~$(A)$ in complex twoplane Grassmannians ${ { {G_2({\mathbb
C}^{m+2})} } }$, which means a
tube over a totally geodesic $G_{2}(\mathbb C^{m+1})$ in
${G_2({\mathbb C}^{m+2})}$, by
the Reeb parallel structure Jacobi operator ${\nabla}_{\xi}R_{\xi}=0$.
Keywords:real hypersurfaces, complex twoplane Grassmannians, Hopf hypersurface, Reeb parallel, structure Jacobi operator Categories:53C40, 53C15 

17. CMB 2013 (vol 57 pp. 546)
 Kalantar, Mehrdad

Compact Operators in Regular LCQ Groups
We show that a regular locally compact quantum group $\mathbb{G}$ is discrete
if and only if $\mathcal{L}^{\infty}(\mathbb{G})$ contains nonzero compact operators on
$\mathcal{L}^{2}(\mathbb{G})$.
As a corollary we classify all discrete quantum groups among
regular locally compact quantum groups $\mathbb{G}$ where
$\mathcal{L}^{1}(\mathbb{G})$ has the RadonNikodym property.
Keywords:locally compact quantum groups, regularity, compact operators Category:46L89 

18. CMB 2012 (vol 57 pp. 166)
19. CMB 2012 (vol 57 pp. 51)
 Fošner, Ajda; Lee, TsiuKwen

Jordan $*$Derivations of FiniteDimensional Semiprime Algebras
In the paper, we characterize Jordan $*$derivations of a $2$torsion
free, finitedimensional semiprime algebra $R$ with involution $*$. To
be precise, we prove the theorem: Let $deltacolon R o R$ be a Jordan
$*$derivation. Then there exists a $*$algebra decomposition
$R=Uoplus V$ such that both $U$ and $V$ are invariant under
$delta$. Moreover, $*$ is the identity map of $U$ and $delta,_U$ is a
derivation, and the Jordan $*$derivation $delta,_V$ is inner.
We also prove the theorem: Let $R$ be a noncommutative, centrally
closed prime algebra with involution $*$, $operatorname{char},R
e 2$,
and let $delta$ be a nonzero Jordan $*$derivation of $R$. If $delta$ is
an elementary operator of $R$, then $operatorname{dim}_CRlt infty$ and
$delta$ is inner.
Keywords:semiprime algebra, involution, (inner) Jordan $*$derivation, elementary operator Categories:16W10, 16N60, 16W25 

20. CMB 2012 (vol 56 pp. 801)
 Oberlin, Richard

Estimates for Compositions of Maximal Operators with Singular Integrals
We prove weaktype $(1,1)$ estimates for compositions of maximal
operators with singular integrals. Our main object of interest is the
operator $\Delta^*\Psi$ where $\Delta^*$ is Bourgain's maximal
multiplier operator and $\Psi$ is the sum of several modulated
singular integrals; here our method yields a significantly improved
bound for the $L^q$ operator norm when $1 \lt q \lt 2.$ We also consider
associated variationnorm estimates.
Keywords:maximal operator calderonzygmund Category:42A45 

21. 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 discalgebra,
\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 calculus Categories:47A10, 47A53, 47A60, 47B07 

22. CMB 2012 (vol 57 pp. 25)
 Bourin, JeanChristophe; Harada, Tetsuo; Lee, EunYoung

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 inequalities Categories:47A63, 15A45 

23. CMB 2012 (vol 56 pp. 503)
 Bu, Qingying

Weak Sequential Completeness of $\mathcal K(X,Y)$
For Banach spaces $X$ and $Y$, we show that if $X^\ast$ and $Y$ are
weakly sequentially complete and every weakly compact operator from
$X$ to $Y$ is compact then the space of all compact operators from $X$
to $Y$ is weakly sequentially complete. The converse is also true if,
in addition, either $X^\ast$ or $Y$ has the bounded compact
approximation property.
Keywords:weak sequential completeness, reflexivity, compact operator space Categories:46B25, 46B28 

24. CMB 2011 (vol 56 pp. 306)
25. CMB 2011 (vol 56 pp. 593)