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Search: MSC category 47A30 ( Norms (inequalities, more than one norm, etc.) )

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1. CMB 2014 (vol 58 pp. 207)

 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 $C^*$-moduleCategories:47A30, 46L05, 46L08, 47B47, 15A60

2. 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

3. 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 operatorCategories:47B47, 47B10, 47A30

4. CMB 2000 (vol 43 pp. 406)

Borwein, David
 Weighted Mean Operators on $l_p$ The weighted mean matrix $M_a$ is the triangular matrix $\{a_k/A_n\}$, where $a_n > 0$ and $A_n := a_1 + a_2 + \cdots + a_n$. It is proved that, subject to $n^c a_n$ being eventually monotonic for each constant $c$ and to the existence of $\alpha := \lim \frac{A_n}{na_n}$, $M_a \in B(l_p)$ for $1 < p < \infty$ if and only if $\alpha < p$. Keywords:weighted means, operators on $l_p$, norm estimatesCategories:47B37, 47A30, 40G05

5. CMB 1999 (vol 42 pp. 87)

 Some norm inequalities for operators Let $A_i$, $B_i$ and $X_i$ $(i=1, 2, \dots, n)$ be operators on a separable Hilbert space. It is shown that if $f$ and $g$ are nonnegative continuous functions on $[0,\infty)$ which satisfy the relation $f(t)g(t) =t$ for all $t$ in $[0,\infty)$, then $$\Biglvert \,\Bigl|\sum^n_{i=1} A^*_i X_i B_i \Bigr|^r \,\Bigrvert^2 \leq \Biglvert \Bigl( \sum^n_{i=1} A^*_i f (|X^*_i|)^2 A_i \Bigr)^r \Bigrvert \, \Biglvert \Bigl( \sum^n_{i=1} B^*_i g (|X_i|)^2 B_i \Bigr)^r \Bigrvert$$ for every $r>0$ and for every unitarily invariant norm. This result improves some known Cauchy-Schwarz type inequalities. Norm inequalities related to the arithmetic-geometric mean inequality and the classical Heinz inequalities are also obtained. Keywords:Unitarily invariant norm, positive operator, arithmetic-geometric mean inequality, Cauchy-Schwarz inequality, Heinz inequalityCategories:47A30, 47B10, 47B15, 47B20
 Simple conditions for matrices to be bounded operators on $l_p$ The two theorems proved yield simple yet reasonably general conditions for triangular matrices to be bounded operators on $l_p$. The theorems are applied to N\"orlund and weighted mean matrices. Keywords:Triangular matrices, NÃ¶rlund matrices, weighted means, operators, on $l_p$.Categories:47B37, 47A30, 40G05