The Brockett-Wegner diagonalizing flow Ḣt = [ Ht, [Ht, A] ] is studied. Global existence and uniqueness of solutions of this evolution equation is proved on the space B[H ] of bounded operators on a complex Hilbert space H . Local existence is proved for certain unbounded initial operators H0. Furthermore, if H0, A are Hilbert-Schmidt operators, it is demonstrated that Ht strongly converges to ...

Motivated by a spectral analysis of the generator completely positive trace-preserving semigroup, we analyze real functionalA,B∈Mn(C)→r(A,B)=12(〈[B,A],BA〉+〈[B,A⁎],BA⁎〉)∈R where 〈A,B〉:=tr(A⁎B) is Hilbert-Schmidt inner product, and [A,B]:=AB−BA commutator. In particular discuss upper lower bounds form c−‖A‖2‖B‖2≤r(A,B)≤c+‖A‖2‖B‖2 ‖A‖ Frobenius norm. We prove that optimal are given c±=1±22. If A r...

In these notes we provide an introduction to compact linear operators on Banach and Hilbert spaces. These operators behave very much like familiar finite dimensional matrices, without necessarily having finite rank. For more thorough treatments, see [RS, Y]. Definition 1 Let X and Y be Banach spaces. A linear operator C : X → Y is said to be compact if for each bounded sequence {xi}i∈IN ⊂ X , t...

Many machine learning algorithms can be formulated in the framework of statistical independence such as the Hilbert Schmidt Independence Criterion. In this paper, we extend this criterion to deal with structured and interdependent observations. This is achieved by modeling the structures using undirected graphical models and comparing the Hilbert space embeddings of distributions. We apply this...

We introduce a Riemannian metric with non positive curvature in the (infinite dimensional) manifold Σ∞ of positive invertible operators of a Hilbert space H, which are scalar perturbations of Hilbert-Schmidt operators. The (minimal) geodesics and the geodesic distance are computed. It is shown that this metric, which is complete, generalizes the well known non positive metric for positive defin...

This paper presents a framework for computing random operator-valued feature maps for operator-valued positive definite kernels. This is a generalization of the random Fourier features for scalar-valued kernels to the operator-valued case. Our general setting is that of operator-valued kernels corresponding to RKHS of functions with values in a Hilbert space. We show that in general, for a give...