Banach Journal of Mathematical Analysis

Geometry of the left action of the p-Schatten groups

Maria Eugenia Di Iorio y Lucero

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Let $\mathcal{H}$ be an infinite dimensional Hilbert space, $\mathcal{B}_{p}\left(\mathcal{H}\right)$ the $p$-Schatten class of $\mathcal{H}$ and $U_{p}\left(\mathcal{H}\right)$ be the Banach-Lie group of unitary operators which are $p$-Schatten perturbations of the identity. Let $A$ be a bounded selfadjoint operator in $\mathcal{H}$. We show that $$\mathcal{O}_A:=\left\{UA : U \in U_{p}\left(\mathcal{H}\right) \right\}$$ is a smooth submanifold of the affine space $A + \mathcal{B}_{p}\left(\mathcal{H}\right)$ if only if $A$ has closed range. Furthermore, it is a homogeneous reductive space of $U_{p}\left(\mathcal{H}\right)$. We introduce two metrics: one via the ambient Finsler metric induced as a submanifold of $A + \mathcal{B}_{p}\left(\mathcal{H}\right)$, the other, by means of the quotient Finsler metric provided by the homogeneous space structure. We show that $\mathcal{O}_A$ is a complete metric space with the rectifiable distance of these metrics.

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Banach J. Math. Anal., Volume 7, Number 1 (2013), 73-87.

First available in Project Euclid: 22 January 2013

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Zentralblatt MATH identifier

Primary: 47B10: Operators belonging to operator ideals (nuclear, p-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20]
Secondary: 46T05: Infinite-dimensional manifolds [See also 53Axx, 57N20, 58Bxx, 58Dxx] 2057N 58B20: Riemannian, Finsler and other geometric structures [See also 53C20, 53C60]

Analytic submanifold Finsler metric Riemannian metric Schatten operator


Di Iorio y Lucero, Maria Eugenia. Geometry of the left action of the p-Schatten groups. Banach J. Math. Anal. 7 (2013), no. 1, 73--87. doi:10.15352/bjma/1358864549.

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