we characterize compact composition operators on real banach spaces of complex-valued bounded lipschitz functions on metric spaces, not necessarily compact, with lipschitz involutions and determine their spectra.

In 1934, H. Whitney asked how one can determine whether a real-valued function on a closed subset of Rn is the restriction of a Cm-function on Rn. A complete answer to this question was found much later by C. Fefferman in the early 2000s. Here, we work in an o-minimal expansion of a real closed field and solve the C1-case of Whitney’s Extension Problem in this context. Our main tool is a defina...

Several classes of hereditarily normal spaces are characterized in terms of extending upper semi-continuous compact-valued mappings. The case of controlled extensions is considered as well. Applications are obtained for real-valued semi-continuous functions.

For any reduced commutative $f$-ring with identity and bounded inversion, we show that a condition which is obviously necessary for the socle of the ring to coincide with the socle of its bounded part, is actually also sufficient. The condition is that every minimal ideal of the ring consist entirely of bounded elements. It is not too stringent, and is satisfied, for instance, by rings of ...

In the paper the existing results concerning a special kind of trajectories and the theory of first return continuous functions connected with them are used to examine some algebraic properties of classes of functions. To that end we define a new class of functions (denoted Conn) contained between the families (widely described in literature) of Darboux Baire 1 functions (DB1) and connectivity ...