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1. CMB Online first
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 |
2. CMB 2012 (vol 57 pp. 42)
Covering the Unit Sphere of Certain Banach Spaces by Sequences of Slices and Balls e prove that, given any covering of any infinite-dimensional 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 point-finite 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 |
3. CMB 2012 (vol 57 pp. 145)
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 calculus Categories:47A10, 47A53, 47A60, 47B07 |
4. CMB 2012 (vol 57 pp. 25)
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 |
5. CMB 2011 (vol 56 pp. 400)
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 algebra Categories:46E22, 47B32, 47L45 |
6. CMB 2011 (vol 54 pp. 498)
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, self-adjoint operators, closed operators, closure of operators Category:47A05 |
7. CMB 2006 (vol 49 pp. 203)
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 well-known result on the existence of the ergodic
Hilbert transform.
Keywords:Hilbert transform, admissible processes Categories:28D05, 37A99 |
8. CMB 2004 (vol 47 pp. 298)
Near Triangularizability Implies Triangularizability In this paper we consider collections of
compact operators on a real or
complex Banach space including linear operators
on finite-dimensional 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 |
9. CMB 1999 (vol 42 pp. 427)
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 theta-functions, Hilbert class fields Categories:33C05, 33E05, 11R20, 11R29 |