Some results on generalized finite operators and range kernel orthogonality in Hilbert spaces

Let ℋ {\mathcal{ {\mathcal H} }} be a complex Hilbert space and ℬ ( ℋ ) {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) denotes the algebra of all bounded linear operators acting on ℋ {\mathcal{ {\mathcal H} }} . In this paper, we present some new pairs of generalized finite operators. More precisely, new pairs of operators ( A , B ) ∈ ℬ ( ℋ ) × ℬ ( ℋ ) \left(A,B)\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\times {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) satisfying: ∥ A X − X B − I ∥ ≥ 1 , for all X ∈ ℬ ( ℋ ) . \parallel AX-XB-I\parallel \ge 1,\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}X\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}). An example under which the class of such operators is not invariant under similarity orbit is given. Range kernel orthogonality of generalized derivation is also studied.

Analysis of non-reversible Markov chains via similarity orbits

In this paper we develop an in-depth analysis of non-reversible Markov chains on denumerable state space from a similarity orbit perspective. In particular, we study the class of Markov chains whose transition kernel is in the similarity orbit of a normal transition kernel, such as that of birth–death chains or reversible Markov chains. We start by identifying a set of sufficient conditions for a Markov chain to belong to the similarity orbit of a birth–death chain. As by-products, we obtain a spectral representation in terms of non-self-adjoint resolutions of identity in the sense of Dunford [21] and offer a detailed analysis on the convergence rate, separation cutoff and L2-cutoff of this class of non-reversible Markov chains. We also look into the problem of estimating the integral functionals from discrete observations for this class. In the last part of this paper we investigate a particular similarity orbit of reversible Markov kernels, which we call the pure birth orbit, and analyse various possibly non-reversible variants of classical birth–death processes in this orbit.

Some Convexity Features Associated with Unitary Orbits

Let be the real linear space of n × n complex Hermitian matrices. The unitary (similarity) orbit of C ∈ is the collection of all matrices unitarily similar to C. We characterize those C ∈ such that every matrix in the convex hull of can be written as the average of two matrices in . The result is used to study spectral properties of submatrices of matrices in , the convexity of images of under linear transformations, and some related questions concerning the joint C-numerical range of Hermitian matrices. Analogous results on real symmetric matrices are also discussed.

GEOMETRY OF CONDITIONAL EXPECTATIONS AND FINITE INDEX

Let e be the Jones projection associated to a conditional expectation [Formula: see text] where [Formula: see text] are von Neumann algebras. We prove that the similarity orbit of e by invertibles of [Formula: see text] is an homogeneous space iff the index of E is finite. If also [Formula: see text], then this orbit is a covering space for the orbit of E.