A new approach to the generalised Brownian motion introduced by M. Bo_ zejko and R. Speicher is described, based on symmetry rather than deformation. The symmetrisation principle is provided by Joyal's notions of tensorial and combinatorial species. Any such species V gives rise to an endofunctor FV of the category of Hilbert spaces with contractions mapping a Hilbert space H to a symmetric Hil...

These notes are organized as follows. We first discuss finite dimensional vector spaces and additional structures that we can impose on them. We then introduce Hilbert spaces, which play a central role in this class, and talk about the subtleties that we have to deal with when working in infinite dimension. Finally, we talk about matrices and linear operators, in the context of linear mappings ...

In this paper, we introduce and study a new system of generalized nonlinear mixed variational inequalities in Hilbert spaces. We prove the existence and uniqueness of solution for this system of generalized nonlinear mixed variational inequalities. We also give the convergence of the Mann iterative sequences with errors for this system of generalized nonlinear mixed variational inequalities in ...

We detect Hilbert manifolds among homogeneous metric spaces and apply the obtained results to recognizing Hilbert manifolds among homogeneous spaces of the form G/H where G is a metrizable topological group and H is a closed balanced subgroup of G.

Let M be a connected compact quantizable Kähler manifold equipped with a Hamiltonian action of a connected compact Lie group G. Let M//G = φ(0)/G = M0 be the symplectic quotient at value 0 of the moment map φ. The space M0 may in general not be smooth. It is known that, as vector spaces, there is a natural isomorphism between the quantum Hilbert space over M0 and the G-invariant subspace of the...

A new approach to the generalised Brownian motion introduced by M. Bożejko and R. Speicher is described, based on symmetry rather than deformation. The symmetrisation principle is provided by Joyal’s notions of tensorial and combinatorial species. Any such species V gives rise to an endofunctor FV of the category of Hilbert spaces with contractions mapping a Hilbert space H to a symmetric Hilbe...

In this paper, we study the split-feasibility problem in Hilbert spaces by using the projected reflected gradient algorithm. As applications, we study the convex linear inverse problem and the split-equality problem in Hilbert spaces, and we give new algorithms for these problems. Finally, numerical results are given for our main results.

A subproduct system of two-dimensional Hilbert spaces can generate an Arveson system of type I1 only. All possible cases are classified up to isomorphism. This work is triggered by a question of Bhat: can a subproduct system of n-dimensional Hilbert spaces generate an Arveson system of type II or III? The question is still open for n > 2.

In this paper, we study an online learning algorithm in Reproducing Kernel Hilbert Spaces (RKHS) and general Hilbert spaces. We present a general form of the stochastic gradient method to minimize a quadratic potential function by an independent identically distributed (i.i.d.) sample sequence, and show a probabilistic upper bound for its convergence.

The notion of g-frames for Hilbert spaces was introduced and studied by Wenchang Sun [16] as a generalization of the notion of frames. In this paper, we define computable g-frames in computable Hilbert spaces and obtain computable versions of some of their characterizations and related results.