Similar to ordinary differential equations, rough paths and equations can be formulated in a Banach space setting. For $\alpha\in (1/3,1/2)$, we give criteria for when approximate space-valued weakly geometric $\alpha$-rough by signatures of curves bounded variation, given some tuning the H\"older parameter. We show that these are satisfied on Hilbert spaces. As an application, obtain Wong-Zaka...

Let X and Y be two Hilbert spaces, and L(X, Y ) the space of bounded linear transformations from X into Y . Let {An} ⊂ L(X, Y ) be a weakly periodic sequence of period T . Spectral theory of weakly periodic sequences in a Hilbert space is studied by H. L. Hurd and V. Mandrekar (1991). In this work we proceed further to characterize {An} by a positive measure μ and a number T of L(X, X)-valued f...

Correlation functions of the XXZ spin chain in the critical regime is studied at zerotemperature. They are exactly represented in the Fredholm determinant form and are related with an operator-valued Riemann–Hilbert problem. Analyzing this problem we prove that a two-point correlation function consisting of sufficiently separated spin operators is expressed by power-functions of the distance be...

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...

In this work we introduce a theory of stochastic integration for operator-valued integrands with respect to some classes cylindrical martingale-valued measures in Hilbert spaces. The integral is constructed via the radonification martingales by Hilbert-Schmidt operator theorem and unifies several other theories particular, our covers space valued L\'{e}vy process (which not required satisfy any...

The aim of this paper is to present a unified framework in the setting Hilbert $$C^*$$ -modules for scalar- and vector-valued reproducing kernel spaces -valued spaces. We investigate conditionally negative definite kernels with values -algebra adjointable operators acting on -module. In addition, we show that there exists two-sided connection between positive -modules. Furthermore, explore some...