We characterize finite codimensional linear isometries on two spaces, C (n)[0; 1] and Lip [0; 1], where C (n)[0; 1] is the Banach space of n-times continuously differentiable functions on [0; 1] and Lip [0; 1] is the Banach space of Lipschitz continuous functions on [0; 1]. We will see they are exactly surjective isometries. Also, we show that C (n)[0; 1] and Lip [0; 1] admit neither isometric shifts nor backward shifts.

We introduce a new class of Banach spaces, called generalized-lush spaces (GL-spaces for short), which contains almost-CL-spaces, separable lush spaces (in particular, separable C-rich subspaces of C(K)), and even the two-dimensional space with hexagonal norm. We find that the space C(K,E) of vector-valued continuous functions is a GL-space whenever E is, and show that the set of GL-spaces is stable under c₀-, l₁- and $l_{\infty }$-sums. As an application, we prove that the Mazur-Ulam property holds for a larger...

Let X and Y be Banach spaces. An operator G: X → Y is a Daugavet center if ‖G +T‖ = ‖G‖+‖T‖ for every rank-1 operator T. For every Daugavet center G we consider a certain set of operators acting from X, so-called G-narrow operators. We prove that if J is the natural embedding of Y into a Banach space E, then E can be equivalently renormed so that an operator T is (J ○ G)-narrow if and only if T is G-narrow. We study G-rich subspaces of X: Z ⊂ X is called G-rich if the quotient map q: X → X/Z is...

Suppose that a real nonatomic function space on $[0,1]$ is equipped with two rearrangement-invariant norms $\parallel ·\parallel$ and $\left|\right||·|\left|\right|$. We study the question whether or not the fact that $(X,\parallel ·\parallel )$ is isometric to $(X,|\left|\right|·\left|\right|\left|\right)$ implies that $\parallel f\parallel =\left|\right|\left|f\right|\left|\right|$ for all $f$ in $X$. We show that in strictly monotone Orlicz and Lorentz spaces this is equivalent to asking whether or not the norms are defined by equal Orlicz functions, respĿorentz weights. We show that the above implication holds true in most rearrangement-invariant spaces, but we also identify a class...

Isometric Sobolev spaces on finite graphs are characterized. The characterization implies that the following analogue of the Banach-Stone theorem is valid: if two Sobolev spaces on 3-connected graphs, with the exponent which is not an even integer, are isometric, then the corresponding graphs are isomorphic. As a corollary it is shown that for each finite group and each p which is not an even integer, there exists n ∈ ℕ and a subspace $L\subset \ell {ⁿ}_p$ whose group of isometries is the direct product × ℤ₂.

We investigate isometric composition operators on the weighted Dirichlet space ${𝒟}_{\alpha }$ with standard weights ${(1-|z|}^2{)}^{\alpha }$, $\alpha >-1$. The main technique used comes from Martín and Vukotić who completely characterized the isometric composition operators on the classical Dirichlet space $𝒟$. We solve some of these but not in general. We also investigate the situation when ${𝒟}_{\alpha }$ is equipped with another equivalent norm.

We prove that some Banach spaces X have the property that every Banach space that can be isometrically embedded in X can be isometrically and linearly embedded in X. We do not know if this is a general property of Banach spaces. As a consequence we characterize for which ordinal numbers α, β there exists an isometric embedding between $C_0(\alpha +1)$ and $C_0(\beta +1)$.

It is shown that $l_2^n$ imbeds isometrically into $l_4^{n^2+1}$ provided that n is a prime power plus one, in the complex case. This and similar imbeddings are constructed using elementary techniques from number theory, combinatorics and coding theory. The imbeddings are related to existence of certain cubature formulas in numerical analysis.

We show that if T is an isometry (as metric spaces) from an open subgroup of the group of invertible elements in a unital semisimple commutative Banach algebra A onto a open subgroup of the group of invertible elements in a unital Banach algebra B, then $T{\left(1\right)}^{-1}T$ is an isometrical group isomorphism. In particular, $T{\left(1\right)}^{-1}T$ extends to an isometrical real algebra isomorphism from A onto B.

We study isometries between spaces of weighted holomorphic functions. We show that such isometries have a canonical form determined by a group of homeomorphisms of a distinguished subset of the range and domain. A number of invariants for these isometries are determined. For specific families of weights we classify the form isometries can take.

A characterization of isometries of complex Musielak-Orlicz spaces $L_{\Phi }$ is given. If $L_{\Phi }$ is not a Hilbert space and $U:L_{\Phi }\to L_{\Phi }$ is a surjective isometry, then there exist a regular set isomorphism τ from (T,Σ,μ) onto itself and a measurable function w such that U(f) = w ·(f ∘ τ) for all $f\in L_{\Phi }$. Isometries of real Nakano spaces, a particular case of Musielak-Orlicz spaces, are also studied.

We improve the Mazur-Ulam theorem by relaxing the surjectivity condition.

Let $X$ and $E$ be a Banach space and a real Banach lattice, respectively, and let $\Gamma$ denote an infinite set. We give concise proofs of the following results: (1) The dual space $X^{*}$ contains an isometric copy of $c_0$ iff $X^{*}$ contains an isometric copy of ${\ell }_{\infty }$, and (2) $E^{*}$ contains a lattice-isometric copy of $c_0\left(\Gamma \right)$ iff $E^{*}$ contains a lattice-isometric copy of ${\ell }_{\infty }\left(\Gamma \right)$.

Let X and Y be Banach spaces. We give a “non-separable” proof of the Kalton-Werner-Lima-Oja theorem that the subspace (X,X) of compact operators forms an M-ideal in the space (X,X) of all continuous linear operators from X to X if and only if X has Kalton’s property (M*) and the metric compact approximation property. Our proof is a quick consequence of two main results. First, we describe how Johnson’s projection P on (X,Y)* applies to f ∈ (X,Y)* when f is represented via a Borel (with respect to...