Let $$\{a_n\}_{n\ge 0}$$ and $$\{b_n\}_{n\ge be sequences of scalars. Suppose $$a_n \ne 0$$ for all $$n \ge . We consider the tridiagonal kernel (also known as band with bandwidth one) $$\begin{aligned} k(z, w) = \sum _{n=0}^\infty ((a_n + b_n z)z^n) \overline{(({a}_n {b}_n {w}) {w}^n)} \qquad (z, w \in \mathbb {D}), \end{aligned}$$ where $$\mathbb {D} \{z {C}: |z| < 1\}$$ Denote by $$M_z$$ mul...

We prove the plasticity of unit ball c. That is, we show that every non-expansive bijection from c onto itself is an isometry. also demonstrate a slightly weaker property for c0 – isometry, provided it has continuous inverse.

We consider the equivalence relation of isometry of separable, complete metric spaces, and show that any equivalence relation induced by a Borel action of a Polish group on a Polish space is Borel reducible to this isometry relation. We also consider the isometry relation restricted to various classes of metric spaces, and produce lower bounds for the complexity in terms of the Borel reducibili...

Every isometry of a finite dimensional euclidean space is a product of reflections and the minimum length of a reflection factorization defines a metric on its full isometry group. In this article we identify the structure of intervals in this metric space by constructing, for each isometry, an explicit combinatorial model encoding all of its minimal length reflection factorizations. The model ...

If f is an isometry, then every distance r > 0 is conserved by f , and vice versa. We can now raise a question whether each mapping that preserves certain distances is an isometry. Indeed, Aleksandrov [1] had raised a question whether a mapping f : X → X preserving a distance r > 0 is an isometry, which is now known to us as the Aleksandrov problem. Without loss of generality, we may assume r =...