In this note, we will discuss how to relate an operator ideal on Banach spaces to the sequential structures it defines. Concrete examples of ideals of compact, weakly compact, completely continuous, Banach-Saks and weakly Banach-Saks operators will be demonstrated.

Let <i>C</i> be a nonempty closed convex subset of a real Hilbert space <i>H</i>. Let <i>S</i> : <i>C</i> <i>C</i> be an asymptotically nonexpansive map in the intermediate sense with the fixed point set <i>F</i>(<i>S</i>). Let <i>A</i> : <i>C</i> <i>H</i> be a Lipschitz continuous map, and <i>VI</i>(<i>C, A</i>) be the set of solutions <i>u</i> <i>C</i> of the variational inequality

Suppose<i>X</i>and<i>Y</i>are locally compact Hausdorff spaces,<i>E</i>and<i>F</i>are Banach spaces, and<i>F</i>is strictly convex. We show that every linear isometry<i>T</i>from<i>C</i>₀(<i>X</i>,<i>E</i>)<i>into</i><i>C</i>₀(<i>Y</i>,<i>F</i>) is essentially a weighted composition operator<i>Tf</i>(<i>y</i>)=<i>h</i>(<i>y</i>)(<i>f</i>((<i>y</i>))).

In this paper, we introduce and study a triple hierarchical variational inequality (THVI) with constraints of minimization and equilibrium problems. More precisely, let Fix ( T ) be the fixed point set of a nonexpansive mapping, let Fix ( T ) be the solution set of a mixed equilibrium problem (MEP), and let <i></i> be the solution set of a minimization problem (MP) for a convex and continuously Frechet differential functional in Hilbert spaces. We want to find a solution Fix ( T ) of a variational inequality with a variational inequality constraint over the intersection of Fix ( T ) , Fix ( T ) , and <i></i>. We propose a hybrid iterative algorithm with regularization to compute approximate solutions of the THVI, and we present the convergence analysis of the proposed iterative algorithm.

We describe the structure of all bijective maps on the cone of positive definite operators acting on a finite and at least two-dimensional complex Hilbert space which preserve the quantum 2 -divergence for some 2 . We prove that any such transformation is necessarily implemented by either a unitary or an antiunitary operator. Similar results concerning maps on the cone of positive semidefinite operators as well as on the set of all density operators are also derived.