In this note self-adjoint realizations of second order elliptic differential expressions with non-local Robin boundary conditions on a domain Ω ⊂ Rn with smooth compact boundary are studied. A Schatten–von Neumann type estimate for the singular values of the difference of the mth powers of the resolvents of two Robin realizations is obtained, and for m > n 2 − 1 it is shown that the resolvent p...

This note represents an approach to the abstract Fredholm theory of trace class (and more generally ~ = {A [ Tr(] A [ ~ ) < oo}) operators on a separable Hi lber t space, ~v{,. There are few new results here but there are a set of new proofs which we feel sheds considerable light on the theory discussed. In part i cular, we would emphasize our proof of Lidskii 's theorem (see Sect. 4): I t was ...

Using properties of an integral transform giving directly the matrix elements of a quantum mechanical operator from the corresponding classical function, we restrict the class of distributions corresponding to bounded operators. As a consequence, we can exhibit a class of functions yielding traceclass operators, and give a bound on their trace-norm.

This is a survey article. We consider different problems in connection with the behavior of functions of operators under perturbations of operators. We deal with three classes of operators: unitary operators, self-adjoint operators, and contractions. We study operator Lipschitz and operator differentiable functions. We also study the behavior of functions under perturbations of an operator by a...

The research of geometric means of matrices is attracting more and more attention from scholars since it has very important applications in mathematics and physics, especially in the von Neuman entropy involving the quantum physics and statistical thermal physical. There are many achievements on the entropy, among which the entropy uncertainty relation, the generalization of the Heisenberg unce...

In this paper we give a constructive characterisation of ultraweakly continuous linear functionals on the space of bounded linear operators on a separable Hilbert space. Let H be a separable complex Hilbert space, with orthonormal basis (en)∞n=1, and B(H) the set of bounded linear operators on H . The weak operator norm associated with the orthonormal basis (en) is defined on B(H) by ‖T ‖w ≡ ∞ ...

in this paper, applying the concept of generalized a-valued norm on a right h  - module and also the notion of ϕ-homomorphism of finsler modules over c  -algebras we rst improve the denition of the finsler module over h  -algebra and then dene ϕ-morphism of finsler modules over h  -algebras. finally we present some results concerning these new ones.

Interpolation inequalities of Gagliardo-Nirenberg type and compactness results for self-adjoint trace-class operators with finite kinetic energy are established. Applying these results to the minimization of various free energy functionals, we determine for instance stationary states of the Hartree problem with temperature corresponding to various statistics. Key-words. Compact self-adjoint ope...

Abstract. Let A and B be almost commuting (i.e., the commutator AB−BA belongs to trace class) self-adjoint operators. We construct a functional calculus φ 7→ φ(A,B) for functions φ in the Besov class B ∞,1(R ). This functional calculus is linear, the operators φ(A,B) and ψ(A,B) almost commute for φ, ψ ∈ B ∞,1(R ), and φ(A,B) = u(A)v(B) whenever φ(s, t) = u(s)v(t). We extend the Helton–Howe trac...

This paper studies the relationship between the singular trace of a weak trace class operator and the asymptotic behaviour of its ζ-function at its leading singularity. For Dixmier measurable and universally measurable operators we describe their measurability in terms of the behaviour of the ζ-function. We use recent advances in singular trace theory and a new approach based on Tauberian theor...