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Manifolds of algebraic elements in the algebra L(H) of bounded linear operators

Jose M. Isidro
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Abstract

Given a complex Hilbert space H, we study the differential geometry of the manifold A of normal algebraic elements in Z=L(H), the algebra of bounded linear operators on H. We represent A as a disjoint union of subsets M of Z and, using the algebraic structure of Z, a torsionfree affine connection $\nabla$ (that is invariant under the group G= Aut (Z) of automorphisms of Z) is defined on each of these connected components and the geodesics are computed. In case M consists of elements that have a fixed finite rank r, (0<r<\infty), G-invariant Riemann and Kähler structures are defined on M which in this way becomes a totally geodesic symmetric holomorphic manifold.

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Jose M. Isidro. 2001-10-30. Manifolds of algebraic elements in the algebra L(H) of bounded linear operators. https://arxiv.org/abs/math/0110315

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