Let M be a Banach C*-module over a C*-algebra A carrying two A-valued inner products 〈., .〉1, 〈., .〉2 which induce equivalent to the given one norms on M. Then the appropriate unital C*-algebras of adjointable bounded A-linear operators on the Hilbert A-modules {M, 〈., .〉1} and {M, 〈., .〉2} are shown to be ∗-isomorphic if and only if there exists a bounded A-linear isomorphism S of these two Hi...

We investigate stochastic Volterra equations and their limiting laws. The we consider are driven by a Hilbert space valued \Levy noise integration kernels may have non-linear dependence on the current state of process. Our method is based an embedding into functions which allows to represent solution equation as boundary value partial differential equation. first gather abstract results give mo...

We analyze the regularized least square algorithm in learning theory with Reproducing Kernel Hilbert Spaces (RKHS). Explicit convergence rates for the regression and binary classification problems are obtained in particular for the polynomial and Gaussian kernels on the n-dimensional sphere and the hypercube. There are two major ingredients in our approach: (i) a law of large numbers for Hilber...

We show there are no non-trivial finite Abelian group-valued measures on the lattice of closed subspaces of an infinite-dimensional Hilbert space, and we use this to establish that the unigroup of the lattice of closed subspaces of an infinite-dimensional Hilbert space is divisible. The main technique is a combinatorial construction of a set of vectors in R2n generalizing properties of those us...

Quaternionic Hilbert spaces play an important role in applied physical sciences especially quantum physics. In this paper, the operator valued frames on quaternionic are introduced and studied. terms of a class partial isometries spaces, parametrization Parseval is obtained. We extend to many properties vector process. Moreover, we show that all can be obtained from single frame. Finally, sever...

We consider an optimal stopping problem for a Hilbert-space valued di usion. We prove that the value function of the problem is the unique viscosity solution of an obstacle problem for the associated parabolic partial di erential equation in the Hilbert space. The results are applied to investigate the pricing of American interest rate options in the lognormal Heath-Jarrow-Morton model of yield...

The aim of this paper is to construct a generalized Fourier analysis for certain Hermitian operators. When A, B are entire functions, then H(A,B) will be the associated reproducing kernel Hilbert spaces of Cn×n-valued functions. In that case, we will construct the expansion theorem forH(A,B) in a comprehensive manner. The spectral functions for the reproducing kernel Hilbert spaces will also be...

The paper contains constructions of Hilbert systems for the action of the circle group T using subgroups of implementable Bogoljubov unitaries w.r.t. Fock representations of the Fermion algebra for suitable data of the selfdual framework: H is the reference Hilbert space, Γ the conjugation and P a basis projection on H. According to a general result for Hilbert systems of this type, the group C...

In this paper, first, we introduce the new concept of (complex) 2–Hilbert spaces, that is, we define the concept of 2–inner product spaces with a complex valued 2–inner product by using the 2–norm. Next, we prove some theorems on Schwartz’s inequality, the polarization identity, the parallelogram laws and related important properties. Finally, we give some open problems related to 2–Hilbert spa...

During the support period July 1, 2011 June 30, 2012, seven research papers were published. They consist of three types: • Research that directly addresses the kernel selection problem in machine learning [1, 2]. • Research that closely relates to the fundamental issues of the proposed research of this grant [3, 4, 5, 6]. • Research that is in the general context of computational mathematics [7...