arXiv subjects
Publications and source records attributed to Konstantinos Konstantos.
A Banach space $X$ has the SHAI property if, for every non-zero Banach space $Y$, every surjective algebra homomorphism from the algebra $\mathcal{L}(X)$ of bounded linear operators on $X$ onto $\mathcal{L}(Y)$ is injective. In this work, we prove that the Bourgain-Rosenthal-Schechtman spaces have the SHAI property, thereby answering a question posed by Johnson, Phillips, and Schechtman in 2022.
For every $1\leq α<ω_1$, we construct an explicit unconditional finite-dimensional decomposition (FDD) $(X_λ)_{λ\in\mathcal{T}_α}$ of the Bourgain-Rosenthal-Schechtman space $R_α^{p,0}$ by blocking its standard martingale difference sequence (MDS) basis. This FDD has strong reproducing properties and supports a theory of distributional representations between the spaces $R_α^{p,0}$, $1\leq α<ω_1$. We use this framework to prove an approximate orthogonal reduction: every bounded linear operator on a limit space $R_α^{p,0}$ is, via a distributional embedding and up to arbitrary precision, reduced to a scalar FDD-diagonal operator. As a consequence, the standard MDS bases of the limit spaces $R_α^{p,0}$ satisfy the factorization property.
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 $I - T$.
For $1<p< \infty$, we show that the Rosenthal $X_{p,w}$ spaces and the Bourgain-Rosenthal-Schechtman $R_ω^{p}$ space have the factorization property and the primary factorization property.