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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 2013 (vol 57 pp. 270)
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 domains Categories:47B47, 47B35, 47L80 |
3. CMB 2011 (vol 55 pp. 646)
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 |
4. CMB 2010 (vol 54 pp. 21)
Generalized D-symmetric Operators II
Let $H$ be a separable,
infinite-dimensional, complex Hilbert space and let $A, B\in{\mathcal L
}(H)$, where ${\mathcal L}(H)$ is the algebra of all bounded linear
operators on $H$. Let $\delta_{AB}\colon {\mathcal L}(H)\rightarrow {\mathcal
L}(H)$ denote the generalized derivation $\delta_{AB}(X)=AX-XB$.
This note will initiate a study on the class of pairs $(A,B)$ such
that $\overline{{\mathcal R}(\delta_{AB})}= \overline{{\mathcal
R}(\delta_{A^{\ast}B^{\ast}})}$.
Keywords:generalized derivation, adjoint, D-symmetric operator, normal operator Categories:47B47, 47B10, 47A30 |
5. CMB 2004 (vol 47 pp. 100)
Invariant Subspaces on $\mathbb{T}^N$ and $\mathbb{R}^N$ Let $N$ be an integer which is larger than one. In this paper we
study invariant subspaces of $L^2 (\mathbb{T}^N)$ under the double
commuting condition. A main result is an $N$-dimensional version of
the theorem proved by Mandrekar and Nakazi. As an application of this
result, we have an $N$-dimensional version of Lax's theorem.
Keywords:invariant subspaces Categories:47A15, 47B47 |