We investigate composition-differentiation operators acting on the Dirichlet space of unit disk. Specifically, we determine characterizations for bounded, compact, and Hilbert-Schmidt operators. In addition, particular classes inducing maps, derive an adjoint formula, compute norm, identify spectrum.

The Hilbert-Schmidt independence criterion (HSIC) is a kernel-based statistical independence measure that can be computed very efficiently. However, it requires us to determine the kernel parameters heuristically because no objective model selection method is available. Least-squares mutual information (LSMI) is another statistical independence measure that is based on direct density-ratio esti...

That is, maps from A to B∗ are given by integral kernels in (A⊗B)∗. However, the validity of this adjunction depends on existence of a genuine tensor product. We recall in an appendix the demonstration that infinitedimensional Hilbert spaces do not have tensor products. Also, we must specify the topology on the duals B∗ and (A⊗B)∗. The strongest conclusion gives these the strong topology, as co...

By introducing the concept of ǫ-convertibility, we extend Nielsen’s and Vidal’s theorems to the entanglement transformation of infinite-dimensional systems. Using an infinitedimensional version of Vidal’s theorem we derive a new stochastic-LOCC (SLOCC) monotone which can be considered as an extension of the Schmidt rank. We show that states with polynomially-damped Schmidt coefficients belong t...