Isometries on Hilbert spaces

Surjective Isometries on Grassmann Spaces

Let H be a complex Hilbert space, n a given positive integer and let Pn(H) be the set of all projections on H with rank n. Under the condition dimH ≥ 4n, we describe the surjective isometries of Pn(H) with respect to the gap metric (the metric induced by the operator norm).

Partial Isometries on Banach Spaces

On Isometries of Euclidean Spaces

Isometries of polyhedral Hilbert geometries

We show that the isometry group of a polyhedral Hilbert geometry coincides with its group of collineations (projectivities) if and only if the polyhedron is not an n-simplex with n ≥ 2. Moreover, we determine the isometry group of the Hilbert geometry on the n-simplex for all n ≥ 2, and find that it has the collineation group as an index-two subgroup. These results confirm, for the class of pol...

On Characterizations of Isometries on Function Spaces

Let X be a Banach space over C. Let T : X → X be an isometric isomorphism (an onto linear operator on X with |Tx| = |x| for all x ∈ X). When X is a Hilbert space, any perturbations of orthonormal basis of X give an isometric isomorphism on X. But, for non-Hilbertian space, one expects that an isometric isomorphism should be something special, and would like to find a characterization for T on s...