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Search: All articles in the CMB digital archive with keyword Hilbert $C^*$-module

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1. CMB Online first

Moslehian, Mohammad Sal; Zamani, Ali
Characterizations of operator Birkhoff--James orthogonality
In this paper, we obtain some characterizations of the (strong) Birkhoff--James orthogonality for elements of Hilbert $C^*$-modules and certain elements of $\mathbb{B}(\mathscr{H})$. Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for $T\in \mathbb{B}(\mathscr{H})$ we prove that if the norm attaining set $\mathbb{M}_T$ is a unit sphere of some finite dimensional subspace $\mathscr{H}_0$ of $\mathscr{H}$ and $\|T\|_{{{\mathscr{H}}_0}^\perp} \lt \|T\|$, then for every $S\in\mathbb{B}(\mathscr{H})$, $T$ is the strong Birkhoff--James orthogonal to $S$ if and only if there exists a unit vector $\xi\in {\mathscr{H}}_0$ such that $\|T\|\xi = |T|\xi$ and $S^*T\xi = 0$. Finally, we introduce a new type of approximate orthogonality and investigate this notion in the setting of inner product $C^*$-modules.

Keywords:Hilbert $C^*$-module, Birkhoff--James orthogonality, strong Birkhoff--James orthogonality, approximate orthogonality
Categories:46L05, 46L08, 46B20

2. 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

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