Finite-codimensional subspaces of Daugavet spaces: projection constants and minimal projections
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908957623189504 |
|---|---|
| author | Kania, Tomasz Lewicki, Grzegorz |
| author_facet | Kania, Tomasz Lewicki, Grzegorz |
| contents | Over the real or complex field, we establish a duality formula for projection constants of finite-codimensional subspaces of Banach spaces with the Daugavet property. If \[ Y=\bigcap_{j=1}^n \ker f_j \subset X, \qquad W=\operatorname{span}\{f_1,\dots,f_n\} \subset X^*, \] then \[ λ(Y,X)=1+λ(W,X^*), \] and minimal projections onto $Y$ correspond exactly to weak$^*$-continuous minimal projections onto $W$. This yields, in particular, a complete description of the hyperplane case: every hyperplane has projection constant $2$, and $\ker f$ admits a minimal projection if and only if $f$ attains its norm.
We then specialise to the real space $X=C[0,1]$. Our second ingredient is a transfer principle from duplication-stable finite-dimensional subspaces of $\ell_1^N$ to piecewise-constant subspaces of $L_1[0,1]\subset M[0,1]=C[0,1]^*$. For the regular symmetric spaces constructed by Chalmers and the second-named author and the second named author and Prophet, respectively, the transferred subspaces retain their projection constants but admit no weak$^*$-continuous minimal projections. Passing to annihilators yields finite-codimensional subspaces of the real space $C[0,1]$ for which the infimum defining the projection constant is not attained.
As a consequence, for every $Λ\in[2,\infty)$ there exists a finite-codimensional subspace $Y$ of the real space $C[0,1]$ such that \[ λ(Y,C[0,1])=Λ, \] and the infimum defining $λ(Y,C[0,1])$ is not attained. For each even codimension $n$ we moreover realise every value in the interval $(2,1+β_n]$, where \[ β_n = \mathsf E_{{\mathsf P}_n}\Bigl|\sum_{j=1}^n \varepsilon_j\Bigr| = n2^{-n}\binom{n}{n/2} \sim \sqrt{\frac{2n}π}, \] $(\varepsilon_j)$ is a Rademacher family on $Ω_n=\{-1,1\}^n$, and $\mathsf{P}_n$ is the uniform probability measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10771 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Finite-codimensional subspaces of Daugavet spaces: projection constants and minimal projections Kania, Tomasz Lewicki, Grzegorz Functional Analysis Primary 41A65, 46B04, Secondary 46B20, 46E15 Over the real or complex field, we establish a duality formula for projection constants of finite-codimensional subspaces of Banach spaces with the Daugavet property. If \[ Y=\bigcap_{j=1}^n \ker f_j \subset X, \qquad W=\operatorname{span}\{f_1,\dots,f_n\} \subset X^*, \] then \[ λ(Y,X)=1+λ(W,X^*), \] and minimal projections onto $Y$ correspond exactly to weak$^*$-continuous minimal projections onto $W$. This yields, in particular, a complete description of the hyperplane case: every hyperplane has projection constant $2$, and $\ker f$ admits a minimal projection if and only if $f$ attains its norm. We then specialise to the real space $X=C[0,1]$. Our second ingredient is a transfer principle from duplication-stable finite-dimensional subspaces of $\ell_1^N$ to piecewise-constant subspaces of $L_1[0,1]\subset M[0,1]=C[0,1]^*$. For the regular symmetric spaces constructed by Chalmers and the second-named author and the second named author and Prophet, respectively, the transferred subspaces retain their projection constants but admit no weak$^*$-continuous minimal projections. Passing to annihilators yields finite-codimensional subspaces of the real space $C[0,1]$ for which the infimum defining the projection constant is not attained. As a consequence, for every $Λ\in[2,\infty)$ there exists a finite-codimensional subspace $Y$ of the real space $C[0,1]$ such that \[ λ(Y,C[0,1])=Λ, \] and the infimum defining $λ(Y,C[0,1])$ is not attained. For each even codimension $n$ we moreover realise every value in the interval $(2,1+β_n]$, where \[ β_n = \mathsf E_{{\mathsf P}_n}\Bigl|\sum_{j=1}^n \varepsilon_j\Bigr| = n2^{-n}\binom{n}{n/2} \sim \sqrt{\frac{2n}π}, \] $(\varepsilon_j)$ is a Rademacher family on $Ω_n=\{-1,1\}^n$, and $\mathsf{P}_n$ is the uniform probability measure. |
| title | Finite-codimensional subspaces of Daugavet spaces: projection constants and minimal projections |
| topic | Functional Analysis Primary 41A65, 46B04, Secondary 46B20, 46E15 |
| url | https://arxiv.org/abs/2604.10771 |