Subspaces of $L^1$ spanned by the even and odd levels of the Haar system
We prove that the closed subspace of $L^1[0,1]$ spanned by the even (respectively, odd) levels of the Haar system is not isomorphic to $L^1[0,1]$.
math.FA↗
arXiv subjects
Thomas Speckhofer
Publications and source records attributed to Thomas Speckhofer.
Factorization in independent sums of Haar system Hardy spaces
We introduce a generalization of the Bourgain-Rosenthal-Schechtman $R_ω^p$ space: Let $Y$ be a Haar system Hardy space, i.e., a separable rearrangement-invariant function space on the unit interval or an associated Hardy space defined via the square function (such as dyadic $H^1$). Then we define $Y_ω$ as the closed linear span in $Y$ of independent distributional copies of the spaces $Y_n$ of dyadic step functions at scale $2^{-n}$. Combining finite-dimensional and infinite-dimensional techniques, we prove that the identity operator $I$ on $Y_ω$ factors through every bounded linear operator $T$ on $Y_ω$ which has large diagonal, and in general, the identity factors either through $T$ or through $I - T$.
math.FA↗
Dimension dependence of factorization problems: Haar system Hardy spaces
For $n\in \mathbb{N}$, let $Y_n$ denote the linear span of the first $n+1$ levels of the Haar system in a Haar system Hardy space $Y$ (this class contains all separable rearrangement-invariant function spaces and also related spaces such as dyadic $H^1$). Let $I_{Y_n}$ denote the identity operator on $Y_n$. We prove the following quantitative factorization result: Fix $Γ,δ,\varepsilon > 0$, and let $n,N \in \mathbb{N}$ be chosen such that $N \ge Cn^2$, where $C = C(Γ,δ,\varepsilon) > 0$ (this amounts to a quasi-polynomial dependence between $\dim Y_N$ and $\dim Y_n$). Then for every linear operator $T\colon Y_N\to Y_N$ with $\|T\|\le Γ$, there exist operators $A,B$ with $\|A\|\|B\|\le 2(1+\varepsilon)$ such that either $I_{Y_n} = ATB$ or $I_{Y_n} = A(I_{Y_N} - T)B$. Moreover, if $T$ has $δ$-large positive diagonal with respect to the Haar system, then we have $I_{Y_n} = ATB$ for some $A,B$ with $\|A\|\|B\|\le (1+\varepsilon)/δ$. If the Haar system is unconditional in $Y$, then an inequality of the form $N \ge Cn$ is sufficient for the above statements to hold (hence, $\dim Y_N$ depends polynomially on $\dim Y_n$). Finally, we prove an analogous result in the case where $T$ has large but not necessarily positive diagonal entries.
math.FA↗
Factorization in Haar system Hardy spaces
A Haar system Hardy space is the completion of the linear span of the Haar system $(h_I)_I$, either under a rearrangement-invariant norm $\|\cdot \|$ or under the associated square function norm \begin{equation*} \Bigl\| \sum_Ia_Ih_I \Bigr\|_{*} = \Bigl\| \Bigl( \sum_I a_I^2 h_I^2 \Bigr)^{1/2} \Bigr\|. \end{equation*} Apart from $L^p$, $1\le p<\infty$, the class of these spaces includes all separable rearrangement-invariant function spaces on $[0,1]$ and also the dyadic Hardy space $H^1$. Using a unified and systematic approach, we prove that a Haar system Hardy space $Y$ with $Y\ne C(Δ)$ ($C(Δ)$ denotes the continuous functions on the Cantor set) has the following properties, which are closely related to the primariness of $Y$: For every bounded linear operator $T$ on $Y$, the identity $I_Y$ factors either through $T$ or through $I_Y - T$, and if $T$ has large diagonal with respect to the Haar system, then the identity factors through $T$. In particular, we obtain that \begin{equation*} \mathcal{M}_Y = \{ T\in \mathcal{B}(Y) : I_Y \ne ATB\text{ for all } A, B\in \mathcal{B}(Y) \} \end{equation*} is the unique maximal ideal of the algebra $\mathcal{B}(Y)$ of bounded linear operators on $Y$. Moreover, we prove similar factorization results for the spaces $\ell^p(Y)$, $1\le p \leq \infty$, and use them to show that they are primary.
math.FA↗