Current work defines Schmidt representation of a bilinear operator T:H1×H2→K, where H1,H2 and K are separable Hilbert spaces. Introducing the concept singular value ordered value, we prove that if T is compact, its values ordered, then has on real We hypothesis existence fundamental.

We study the stability properties of nonlinear multi-task regression in reproducing Hilbert spaces with operator-valued kernels. Such kernels, a.k.a. multi-task kernels, are appropriate for learning problems with nonscalar outputs like multi-task learning and structured output prediction. We show that multi-task kernel regression algorithms are uniformly stable in the general case of infinite-d...

‎Some improvements of Young inequality and its reverse for positive‎ ‎numbers with Kantorovich constant $K(t‎, ‎2)=frac{(1+t)^2}{4t}$‎ ‎are given‎. ‎Using these inequalities some operator inequalities and‎ ‎Hilbert-Schmidt norm versions for matrices are proved‎. ‎In‎ ‎particular‎, ‎it is shown that if $a‎, ‎b$ are positive numbers and‎ ‎$0 leqslant nu leqslant 1,$ then for all integers $ kgeqsl...

We prove the H-infinity error bounds for Lyapunov balanced truncation and for optimal Hankel norm approximation under the assumption that the Hankel operator is nuclear. This is an improvement of the result from Glover, Curtain, and Partington [SIAM J. Control Optim., 26 (1998), pp. 863–898], where additional assumptions were made. The proof is based on convergence of the Schmidt pairs of the H...

Given a self-adjoint operator H0, a self-adjoint trace class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and co-kernel, using limiting absorption principle an explicit set of full Lebesgue measure Λ(H0, F) ⊂ R is defined, such that for all points of this set the wave and the scattering matrices can be defined unambiguously. Many well-known properties of the wave and sc...

Given a self-adjoint operator H 0 , a self-adjoint trace class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and co-kernel, using limiting absorption principle an explicit set of full Lebesgue measure Λ(H 0 , F) ⊂ R is defined, such that for all points of this set the wave and the scattering matrices can be defined unambiguously. Many well-known properties of the wave an...

The linearized collision operator of the Boltzmann equation can in a natural way be written as sum positive multiplication operator, frequency, and an integral operator. Compactness for monatomic single species is classical result, while corresponding result mixtures more recently obtained. In this work compactness polyatomic species, where polyatomicity modeled by discrete internal energy vari...

The aim of this paper is to give a precise proof the completeness Lamb modes and associated modes. This relatively simple short but relies on two powerful mathematical theorems. first one theorem elliptic systems with parameter due Agranovich Vishik. second Locker which gives criterion show set generalized eigenvectors Hilbert-Schmidt discrete operator.

In this note we study some basic properties of the γinterpolation method introduced in [6] e.g. interpolation of Bochner spaces and interpolation of analytic operator valued functions in the sense of Stein. As applications we consider the interpolation of almost summing operators (in particular γ-radonifying and Hilbert-Schmidt operators) and of γ-Sobolev spaces as introduced in [6]. We also co...

We describe a method to investigate the mode shapes in turbulent pipe flow at a given wavenumber pair that are most responsive to harmonic forcing in the sense that the they correspond to the largest singular value in a Schmidt decomposition of the linear NavierStokes operator using the turbulent mean profile as the base flow. The ideas follow logically from the work of Sharma & McKeon (2009), ...