Related papers: Higher derivatives of operator functions in ideals…

In [10], Halmos proved an interesting result that the set of irreducible operators is dense in $\mathcal B(\mathcal H)$ in the sense of Hilbert-Schmidt approximation. In a von Neumann algebra $\mathcal M$ with separable predual, an operator…

Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful normal semi-finite trace $\tau$ and let $S_0(\tau)$ be the algebra of all $\tau$-compact operators affiliated with $\mathcal{M}$. Let $E(\tau)\subseteq S_0(\tau)$ be a…

Let $\mathcal{M}$ be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space $H$ equipped with a semifinite faithful normal…

Fix a unital $C^*$-algebra $\mathscr{A}$, and write $\mathscr{A}_{sa}$ for the set of self-adjoint elements of $\mathscr{A}$. Also, if $f:\mathbb{R}\to\mathbb{C}$ is a continuous function, then write…

In this article, we aim to provide a satisfactory algebraic description of the set of affiliated operators for von Neumann algebras. Let $\mathscr{M}$ be a von Neumann algebra acting on a Hilbert space $\mathcal{H}$, and let…

The class of absolutely norming operators on complex Hilbert spaces of arbitrary dimensions was introduced in [6] and a spectral characterization theorem for these operators was established in [11]. In this paper we extend the concept of…

Let $\mathcal{M}$ be a von Neumann algebra, $\mathcal{I}$ a weak-operator dense ideal in $\mathcal{M}$, and $\Phi$ a unitarily invariant $\|\cdot\|$-dominating norm on $\mathcal{I}$. In this paper, we provide a necessary and sufficient…

Given a von Neumann algebra $M$ denote by $S(M)$ and $LS(M)$ respectively the algebras of all measurable and locally measurable operators affiliated with $M.$ For a faithful normal semi-finite trace $\tau$ on $M$ let $S(M, \tau)$ (resp.…

For a normal measurable operator $a$ affiliated with a von Neumann factor $\mathcal{M}$ we show: If $\mathcal{M}$ is infinite, then there is $\lambda_0\in \mathbb{C}$ so that for $\varepsilon>0$ there are…

In this paper we introduce a new technique for proving norm inequalities in operator ideals with an unitarily invariant norm. Among the well known inequalities which can be proved with this technique are the L\"owner-Heinz inequality,…

Let $\mathscr{M}$ be a $II_1$ factor acting on the Hilbert space $\mathscr{H}$, and $\mathscr{M}_{\textrm{aff}}$ be the Murray-von Neumann algebra of closed densely-defined operators affiliated with $\mathscr{M}$. Let $\tau$ denote the…

Let ${\mathfrak A}$ be a $C^*$-algebra, $T$ be a locally compact Hausdorff space equipped with a probability measure $P$ and let $(A_t)_{t\in T}$ be a continuous field of operators in ${\mathfrak A}$ such that the function $t \mapsto A_t$…

Consider a Banach space valued measurable function $f$ and an operator $u$ from the space where {$f$} takes values. If $f $ is Pettis integrable, a classical result due to J. Diestel shows that composing it with $u$ gives a Bochner…

We prove a general criterion for a von Neumann algebra $M$ in order to be in standard form. It is formulated in terms of an everywhere defined, invertible, antilinear, a priori not necessarily bounded operator, intertwining $M$ with its…

We study a specific family of symmetric norms on the algebra $\mathcal B(\mathcal H)$ of operators on a separable infinite-dimensional Hilbert space. With respect to each symmetric norm in this family the identity operator fails to attain…

We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fr\'echet…

This article investigates the soft-interior and the soft-cover of operator ideals. These operations, and especially the first one, have been widely used before, but making their role explicit and analyzing their interplay with the…

This is a continuation of our paper \cite{AP2}. We prove that for functions $f$ in the H\"older class $\L_\a(\R)$ and $1<p<\be$, the operator $f(A)-f(B)$ belongs to $\bS_{p/\a}$, whenever $A$ and $B$ are self-adjoint operators with…

We establish the following results on higher order $\mathcal{S}^p$-differentiability, $1<p<\infty$, of the operator function arising from a continuous scalar function $f$ and self-adjoint operators defined on a fixed separable Hilbert…

Let $\Phi:TM\to TM$ be a positive-semidefinite symmetric operator of class $C^1$ defined on a complete non-compact manifold $M$ isometrically immersed in a Hadamard space $\bar{M}$. In this paper, we given conditions on the operator $\Phi$…