Refinements of the Blanco-Koldobsky-Turnšek Theorem

We refine the well-known Blanco-Koldobsky-Turnšek Theorem which states that a norm one linear operator defined on a Banach space is an isometry if and only if it preserves orthogonality at every element of the space. We improve the result for Banach spaces in which the set of all smooth points forms a dense $G_δ$-set by proving that a norm one linear operator that preserves orthogonality on a dense subset of the space is an isometry. We further demonstrate that if such an operator preserves orthogonality on a hyperplane not passing through the origin then it is an isometry. In the context of finite-dimensional Banach spaces, we prove that preserving orthogonality on the set all extreme points of the unit ball forces the operator to be an isometry, which substantially refines Blanco-Koldobsky-Turnšek theorem. Finally, for finite-dimensional polyhedral spaces, we establish the significance of the set of all $k$-smooth points for any possible $k,$ in the study of isometric theory.

💡 Research Summary

The paper revisits the classical Blanco‑Koldobsky‑Turnšek theorem, which characterizes norm‑one linear operators on a Banach space as isometries precisely when they preserve Birkhoff‑James orthogonality at every point. The authors introduce the notion of a K‑set (Koldobsky‑set): a subset A of the unit sphere such that any norm‑one operator preserving Birkhoff‑James orthogonality at each point of A must be a scalar multiple of an isometry. Their aim is to identify proper K‑sets that are strictly smaller than the whole sphere, thereby refining the original theorem.

The first major contribution concerns Banach spaces whose smooth points form a dense Gδ set. Using the machinery of norm derivatives (ρ′± and the symmetric derivative ρ′) and the geometry of smooth points, they prove several auxiliary results: (i) if an operator preserves orthogonality at a point x and Tx is smooth, then x must be smooth; (ii) preservation on an arbitrary set A automatically extends to A∩Sm X, where Sm X denotes the smooth points of the unit sphere; (iii) the set of points where orthogonality is preserved is closed within Sm X. These observations lead to Lemma 2.4, which guarantees that smooth points remain dense after removing the kernel of a non‑zero operator and that, for a bijection, the intersection Sm X∩T⁻¹(Sm Y) is dense.

Theorem 2.5 shows that in such spaces, an operator that preserves Birkhoff‑James orthogonality on all smooth points must be injective. The proof uses a contradiction argument: assuming a non‑trivial kernel, one constructs a smooth vector orthogonal to a kernel element, which forces a violation of the orthogonality‑preserving property.

The central result, Theorem 2.7, states that if X and Y both have dense Gδ sets of smooth points and a bijective operator T preserves orthogonality on the dense set D = Sm X ∩ T⁻¹(Sm Y), then T is a scalar multiple of an isometry. The proof defines f(x)=‖Tx‖‖x‖, shows that f is constant on D by differentiating along arbitrary directions using the norm derivative, and then extends constancy to the whole space because D is dense and f is continuous. Consequently, ‖Tx‖ = λ‖x‖ for all x, i.e., T = λU with U an isometry.

Corollary 2.8 extracts the immediate refinement: the set Sm X∩S_X (smooth points of the unit sphere) is a K‑set. Theorem 2.9 and Corollary 2.10 further relax the hypothesis: it suffices that orthogonality be preserved on any dense subset U of X, or on a hyperplane H that does not pass through the origin. The latter remark emphasizes that a hyperplane through the origin cannot serve as a K‑set because its inclusion in the kernel of a non‑zero operator would break the isometry conclusion.

In the final part, the authors turn to finite‑dimensional polyhedral Banach spaces. They recall that a point is k‑smooth (order k) iff it lies in the relative interior of an (n−k)-face of the unit ball. Using this characterization, they prove that for any possible k, the set of k‑smooth points is a K‑set. This generalizes earlier results that only extreme points (0‑smooth) form a K‑set in polyhedral spaces.

Overall, the paper systematically expands the scope of the Blanco‑Koldobsky‑Turnšek theorem. By identifying dense smooth subsets, dense hyperplanes, and k‑smooth point sets as K‑sets, the authors demonstrate that global orthogonality preservation is not necessary; local preservation on geometrically significant subsets already forces an operator to be a scalar multiple of an isometry. The work deepens the interplay between norm derivatives, Birkhoff‑James orthogonality, and the fine geometry of Banach spaces, offering new tools for the study of isometries and for applications where only partial orthogonality information is available.