Richard Lechner’s research while affiliated with Johannes Kepler University of Linz and other places

Let (hI)$(h_I)$ denote the standard Haar system on [0, 1], indexed by I∈D$I\in \mathcal {D}$, the set of dyadic intervals and hI⊗hJ$h_I\otimes h_J$ denote the tensor product (s,t)↦hI(s)hJ(t)$(s,t)\mapsto h_I(s) h_J(t)$, I,J∈D$I,J\in \mathcal {D}$. We consider a class of two-parameter function spaces which are completions of the linear span V(δ2)$\mathcal {V}(\delta ^2)$ of hI⊗hJ$h_I\otimes h_J$, I,J∈D$I,J\in \mathcal {D}$. This class contains all the spaces of the form X(Y), where X and Y are either the Lebesgue spaces Lp[0,1]$L^p[0,1]$ or the Hardy spaces Hp[0,1]$H^p[0,1]$, 1≤p<∞$1\le p < \infty$. We say that D:X(Y)→X(Y)$D:X(Y)\rightarrow X(Y)$ is a Haar multiplier if D(hI⊗hJ)=dI,JhI⊗hJ$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$, where dI,J∈R$d_{I,J}\in \mathbb {R}$, and ask which more elementary operators factor through D. A decisive role is played by the Capon projectionC:V(δ2)→V(δ2)$\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)$ given by ChI⊗hJ=hI⊗hJ$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$ if |I|≤|J|$|I|\le |J|$, and ChI⊗hJ=0$\mathcal {C} h_I\otimes h_J = 0$ if |I|>|J|$|I| > |J|$, as our main result highlights: Given any bounded Haar multiplier D:X(Y)→X(Y)$D:X(Y)\rightarrow X(Y)$, there exist λ,μ∈R$\lambda ,\mu \in \mathbb {R}$ such that λC+μ(Id-C)approximately 1-projectionally factors throughD,$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C})\text { approximately 1-projectionally factors through }D, \end{aligned}$i.e., for all η>0$\eta > 0$, there exist bounded operators A, B so that AB is the identity operator Id${{\,\textrm{Id}\,}}$, ‖A‖·‖B‖=1$\Vert A\Vert \cdot \Vert B\Vert = 1$ and ‖λC+μ(Id-C)-ADB‖<η$\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C}) - ADB\Vert < \eta$. Additionally, if C$\mathcal {C}$ is unbounded on X(Y), then λ=μ$\lambda = \mu$ and then Id${{\,\textrm{Id}\,}}$ either factors through D or Id-D${{\,\textrm{Id}\,}}-D$.

Stopping-time Banach spaces is a collective term for the class of spaces of eventually null integrable processes that are defined in terms of the behaviour of the stopping times with respect to some fixed filtration. From the point of view of Banach space theory, these spaces in many regards resemble the classical spaces such as L1 or C(Δ), but unlike these, they do have unconditional bases. In the present paper, we study the canonical bases in the stopping-time spaces in relation to factorising the identity operator thereon. Since we work exclusively with the dyadic-tree filtration, this setup enables us to work with tree-indexed bases rather than directly with stochastic processes. En route to the factorisation results, we develop general criteria that allow one to deduce the uniqueness of the maximal ideal in the algebra of operators on a Banach space. These criteria are applicable to many classical Banach spaces such as (mixed-norm) Lp-spaces, BMO, SL∞, and others.

The classical Banach space $L_1(L_p)$ consists of measurable scalar functions f on the unit square for which \begin{align*}\|f\| = \int_0^1\Big(\int_0^1 |f(x,y)|^p dy\Big)^{1/p}dx < \infty.\end{align*} We show that $L_1(L_p) (1 < p < \infty )$ is primary, meaning that whenever $L_1(L_p) = E\oplus F$ , where E and F are closed subspaces of $L_1(L_p)$ , then either E or F is isomorphic to $L_1(L_p)$ . More generally, we show that $L_1(X)$ is primary for a large class of rearrangement-invariant Banach function spaces.

Stopping-time Banach spaces is a collective term for the class of spaces of eventually null integrable processes that are defined in terms of the behaviour of the stopping times with respect to some fixed filtration. From the point of view of Banach space theory, these spaces in many regards resemble the classical spaces such as $L^1$ or $C(\Delta)$, but unlike these, they do have unconditional bases. In the present paper we study the canonical bases in the stopping-time spaces in relation to factorising the identity operator thereon. Since we work exclusively with the dyadic-tree filtration, this set-up enables us to work with tree-indexed bases rather than directly with stochastic processes. \emph{En route} to the factorisation results, we develop general criteria that allow one to deduce the uniqueness of the maximal ideal in the algebra of operators on a Banach space. These criteria are applicable to many classical Banach spaces such as (mixed-norm) $L^p$-spaces, BMO, $\mathrm{SL^\infty}$ and others.

The classical Banach space $L_1(L_p)$ consists of measurable scalar functions f on the unit square for which $\|f\| = \int_0^1\Big(\int_0^1 |f(x,y)|^p dy\Big)^{1/p}dx < \infty.$ We show that $L_1(L_p)$ $(1 < p < \infty)$ is primary, meaning that, whenever $L_1(L_p) = E\oplus F$ then either E or F is isomorphic to $L_1(L_p)$. More generally we show that $L_1(X)$ is primary, for a large class of rearrangement invariant Banach function spaces.

In this paper we consider the following problem: let Xk, be a Banach space with a normalised basis (e(k, j))j, whose biorthogonals are denoted by ${(e_{(k,j)}^*)_j}$ , for $k\in\N$ , let Z=\ell^\infty(X_k:k\kin\N) be their l∞-sum, and let $T:Z\to Z$ be a bounded linear operator with a large diagonal, i.e., \begin{align*}\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.\end{align*} Under which condition does the identity on Z factor through T? The purpose of this paper is to formulate general conditions for which the answer is positive.

We show that every subsymmetric Schauder basis $(e_j)$ of a Banach space X has the factorization property, i.e. $I_X$ factors through every bounded operator $T\colon X\to X$ with a $\delta$-large diagonal (that is $\inf_j |\langle Te_j, e_j^*\rangle| \geq \delta > 0$, where the $(e_j^*)$ are the biorthogonal functionals to $(e_j)$). Even if X is a non-separable dual space with a subsymmetric weak$^*$ Schauder basis $(e_j)$, we prove that if $(e_j)$ is non-$\ell^1$-splicing (there is no disjointly supported $\ell^1$-sequence in X), then $(e_j)$ has the factorization property. The same is true for $\ell^p$-direct sums of such Banach spaces for all $1\leq p\leq \infty$. Moreover, we find a condition for an unconditional basis $(e_j)_{j=1}^n$ of a Banach space $X_n$ in terms of the quantities $\|e_1+\ldots+e_n\|$ and $\|e_1^*+\ldots+e_n^*\|$ under which an operator $T\colon X_n\to X_n$ with $\delta$-large diagonal can be inverted when restricted to $X_\sigma = [e_j : j\in\sigma]$ for a "large" set $\sigma\subset \{1,\ldots,n\}$ (restricted invertibility of T; see Bourgain and Tzafriri [Israel J. Math. 1987, London Math. Soc. Lecture Note Ser. 1989). We then apply this result to subsymmetric bases to obtain that operators T with a $\delta$-large diagonal defined on any space $X_n$ with a subsymmetric basis $(e_j)$ can be inverted on $X_\sigma$ for some $\sigma$ with $|\sigma|\geq c n^{1/4}$.

We introduce the concept of strategically reproducible bases in Banach spaces and show that operators which have large diagonal with respect to strategically reproducible bases are factors of the identity. We give several examples of classical Banach spaces in which the Haar system is strategically reproducible: multi-parameter Lebesgue spaces, mixed-norm Hardy spaces and most significantly the space L1. Moreover, we show the strategical reproducibility is inherited by unconditional sums.