‎In this paper we try to extend geometric concepts in the context of operator valued tensors‎. ‎To this end‎, ‎we aim to replace the field of scalars $ mathbb{R} $ by self-adjoint elements of a commutative $ C^star $-algebra‎, ‎and reach an appropriate generalization of geometrical concepts on manifolds‎. ‎First‎, ‎we put forward the concept of operator-valued tensors and extend semi-Riemannian...

The aim of this article is to give the proof-theoretic analysis of various subsystems of Feferman's theory T 1 for explicit mathematics which contain the non-constructive-operator and join. We make use of standard proof-theoretic techniques such as cut-elimination of appropriate semi-formal systems and asymmetrical interpretations in standard structures for explicit mathematics.

If T = {T (t); t ≥ 0} is a strongly continuous family of bounded linear operators between two Banach spaces X and Y and f ∈ L1(0, b, X), the convolution of T with f is defined by (T ∗f)(t) = ∫ t 0 T (s)f(t− s)ds. It is shown that T ∗ f is continuously differentiable for all f ∈ C(0, b, X) if and only if T is of bounded semi-variation on [0, b]. Further T ∗ f is continuously differentiable for a...

The theory of anti-selfdual (ASD) Lagrangians developed in [8] allows a variational resolution for equations of the form Λu+Au+∂φ(u)+f = 0 where φ is a convex lower-semi-continuous function on a reflexive Banach space X, f ∈ X∗, A : D(A) ⊂ X → X∗ is a positive linear operator and where Λ : D(Λ) ⊂ X → X∗ is a non-linear operator that satisfies suitable continuity and anti-symmetry properties. AS...

In this paper, we always assume that H is an N-dimensional complex Hilbert space. Let L(H) be the set of all linear operators from H to H . A state ρ of some quantum system, described by H , is a positive semi-definite operator of trace one, in particular, for each unit vector |ψ〉 ∈ H , the operator ρ = |ψ〉〈ψ| is said to be a pure state. The set of all states onH is denoted by D(H). If X,Y ∈ L(...

in this paper, we find explicit solution to the operator equation$txs^* -sx^*t^*=a$ in the general setting of the adjointable operators between hilbert $c^*$-modules, when$t,s$ have closed ranges and $s$ is a self adjoint operator.

We first show that a bounded linear operator $ T $ on a real Banach space $ E $ is quasicompact (Riesz, respectively) if and only if $T': E_{mathbb{C}}longrightarrow E_{mathbb{C}}$ is quasicompact  (Riesz, respectively), where the complex Banach space $E_{mathbb{C}}$ is a suitable complexification of $E$ and $T'$ is the complex linear operator on $E_{mathbb{C}}$ associated with $T$. Next, we pr...

Motivated by the theory of kinetic models in gas dynamics, we obtain an integral representation of lower semi-continuous functions on Rd, d ≥ 1. We use the representation to study the problem of compactness of a family of the solutions of the discrete time BGK model for the compressible Euler equations.