In this paper, we obtain several new continuous selection theorems for multi-valued mappings on completely regular spaces and fixed point theorems for multi-valued maps on nonmetrizable spaces. They, in particular, provide a partial solution of a conjecture of X. Wu.

Abstract Let T: V W be a surjective real linear isometry between full real Hilbert C-modules over real C-algebras A and B, respectively. We show that the following conditions are equivalent:(a) T is a 2-isometry;(b) T is a complete isometry;(c) T preserves ternary products;(d) T preserves inner products;(e) T is a module map. When A and B are commutative, we give a full description of the structure of T.

For each real α,0≤α<1, we give examples of endomorphisms in dimension one with infinite topological entropy which are α‑Hölder; and for each real p,1≤p<∞, we also give examples of endomorphisms in dimension one with infinite topological entropy which are (1,p)-Sobolev. These examples are constructed within a family of endomorphisms with infinite topological entropy and which traverse all α-Hölder and (1,p)-Sobolev classes. Finally, we also give examples of endomorphisms, also in dimension one, which lie in the big and little Zygmund classes, answering a question of M. Benedicks.

The SOC-monotone function (respectively, SOC-convex function) is a scalar valued function that induces a map to preserve the monotone order (respectively, the convex order), when imposed on the spectral factorization of vectors associated with second-order cones (SOCs) in general Hilbert spaces. In this paper, we provide the sufficient and necessary characterizations for the two classes of functions, and particularly establish that the set of continuous SOC-monotone (respectively, SOC-convex) functions coincides with that of continuous matrix monotone (respectively, matrix convex) functions of order 2.

In this paper we study the distribution of scattering resonances for a multi-dimensional semi-classical Schrödinger operator, associated to a potential well in an island at energies close to the maximal one that limits the separation of the well and the surrounding sea.