Left invariant metrics induced by the p-norms of the trace in the matrix algebra are studied on the general lineal group. By means of the Euler-Lagrange equations, existence and uniqueness of extremal paths for the length functional are established, and regularity properties of these extremal paths are obtained. Minimizing paths in the group are shown to have a velocity with constant singular values and multiplicity. In several special cases, these geodesic paths are computed explicitly. In particular the Riemannian geodesics, corresponding to the case p = 2, are characterized as the product of two one-parameter groups. It is also shown that geodesics are one-parameter groups if and only if the initial velocity is a normal matrix. These results are further extended to the context of compact operators with p-summable spectrum, where a differential equation for the spectral projections of the velocity vector of an extremal path is obtained. 1 * 2010 MSC. Primary 58B20, 53C22; Secondary 47D03, 70H03, 70H05. 1
We study the existence and characterization properties of compact Hermitian operators C on a separable Hilbert space H such thatwhere D(K(H) h ) denotes the space of compact real diagonal operators in a fixed base of H and . is the operator norm. We also exhibit a positive trace class operator that fails to attain the minimum in a compact diagonal.Where the suffix ah refers to the anti-Hermitian operators (analogously, the suffix h refers to Hermitian operators). If x ∈ T b (O A ), the existence of a (not necessarily unique) minimal element z 0 such that x = z 0 = inf z : z ∈ K(H) ah , zb − bz = x allows the description of minimal length curves of the manifold by the parametrizationThese z 0 can be described as i(C + D), with C ∈ K(H) h and D a real diagonal operator in the orthonormal base of eigenvectors of A.If we consider a von Neumann algebra A and a von Neumann subalgebra, named B, of A, it has been proved in [6] that for each a ∈ A there always exists a minimal element b 0 in B. It means that Date: September 9, 2018.
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