1. CMB 2016 (vol 59 pp. 769)
 GarcíaPacheco, Francisco Javier; Hill, Justin R.

Geometric Characterizations of Hilbert Spaces
We study some geometric properties related to the set $\Pi_X:=
\{
(x,x^*
)\in\mathsf{S}_X\times \mathsf{S}_{X^*}:x^*
(x
)=1
\}$ obtaining two characterizations of Hilbert spaces
in the category of Banach spaces. We also compute the distance
of a generic element $
(h,k
)\in H\oplus_2 H$ to $\Pi_H$ for $H$ a Hilbert space.
Keywords:Hilbert space, extreme point, smooth, $\mathsf{L}^2$summands Categories:46B20, 46C05 

2. CMB 2016 (vol 59 pp. 497)
3. CMB 2015 (vol 58 pp. 787)
 Kitabeppu, Yu; Lakzian, Sajjad

Nonbranching RCD$(0,N)$ Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups
In this paper, we generalize the finite generation result of
Sormani
to nonbranching $RCD(0,N)$
geodesic spaces (and in particular, Alexandrov spaces) with full
support measures. This is a special case of the Milnor's Conjecture
for complete noncompact $RCD(0,N)$ spaces. One of the key tools
we use is the AbreschGromoll type excess estimates for nonsmooth
spaces obtained by GigliMosconi.
Keywords:Milnor conjecture, non negative Ricci curvature, curvature dimension condition, finitely generated, fundamental group, infinitesimally Hilbertian Categories:53C23, 30L99 

4. 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 

5. CMB 2012 (vol 57 pp. 42)
 Fonf, Vladimir P.; Zanco, Clemente

Covering the Unit Sphere of Certain Banach Spaces by Sequences of Slices and Balls
e prove that, given any covering of any infinitedimensional Hilbert space $H$ by countably many closed balls, some point exists in $H$ which belongs to infinitely many balls. We do that by characterizing isomorphically polyhedral separable Banach spaces as those whose unit sphere admits a pointfinite covering by the union of countably many slices of the unit ball.
Keywords:point finite coverings, slices, polyhedral spaces, Hilbert spaces Categories:46B20, 46C05, 52C17 

6. 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 

7. 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 

8. CMB 2011 (vol 56 pp. 400)
9. CMB 2011 (vol 54 pp. 498)
 Mortad, Mohammed Hichem

On the Adjoint and the Closure of the Sum of Two Unbounded Operators
We prove, under some conditions on the domains, that the adjoint of
the sum of two unbounded operators is the sum of their adjoints in
both Hilbert and Banach space settings. A similar result about the
closure of operators is also proved. Some interesting consequences
and examples "spice up" the paper.
Keywords:unbounded operators, sum and products of operators, Hilbert and Banach adjoints, selfadjoint operators, closed operators, closure of operators Category:47A05 

10. CMB 2006 (vol 49 pp. 203)
 Çömez, Doğan

The Ergodic Hilbert Transform for Admissible Processes
It is shown that the ergodic Hilbert transform
exists for a class of bounded symmetric admissible processes
relative to invertible measure preserving transformations. This
generalizes the wellknown result on the existence of the ergodic
Hilbert transform.
Keywords:Hilbert transform, admissible processes Categories:28D05, 37A99 

11. CMB 2004 (vol 47 pp. 298)
 Yahaghi, Bamdad R.

Near Triangularizability Implies Triangularizability
In this paper we consider collections of
compact operators on a real or
complex Banach space including linear operators
on finitedimensional vector spaces. We show
that such a collection is simultaneously
triangularizable if and only if it is arbitrarily
close to a simultaneously triangularizable
collection of compact operators. As an application
of these results we obtain an invariant subspace
theorem for certain bounded operators. We
further prove that in finite dimensions near
reducibility implies reducibility whenever
the ground field is $\BR$ or $\BC$.
Keywords:Linear transformation, Compact operator,, Triangularizability, Banach space, Hilbert, space Categories:47A15, 47D03, 20M20 

12. CMB 1999 (vol 42 pp. 427)
 Berndt, Bruce C.; Chan, Heng Huat

Ramanujan and the Modular $j$Invariant
A new infinite product $t_n$ was introduced by S.~Ramanujan on the
last page of his third notebook. In this paper, we prove
Ramanujan's assertions about $t_n$ by establishing new connections
between the modular $j$invariant and Ramanujan's cubic theory of
elliptic functions to alternative bases. We also show that for
certain integers $n$, $t_n$ generates the Hilbert class field of
$\mathbb{Q} (\sqrt{n})$. This shows that $t_n$ is a new class
invariant according to H.~Weber's definition of class invariants.
Keywords:modular functions, the Borweins' cubic thetafunctions, Hilbert class fields Categories:33C05, 33E05, 11R20, 11R29 
