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Register a new userFactorization in $SL^infty$
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Authors
Richard Lechner
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We show that the non-separable Banach space $SL^infty$ is primary. This is achieved by directly solving the infinite dimensional factorization problem in $SL^infty$. In particular, we bypass Bourgains localization method.
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In this paper we consider the following problem: Let $X_k$, be a Banach space with a normalized basis $(e_{(k,j)})_j$, whose biorthogonals are denoted by $(e_{(k,j)}^*)_j$, for $kinmathbb{N}$, let $Z=ell^infty(X_k:kinmathbb{N})$ be their $ell^infty$-sum, and let $T:Zto Z$ be a bounded linear operator, with a large diagonal, i.e. $$inf_{k,j} big|e^*_{(k,j)}(T(e_{(k,j)})big|>0.$$ 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 classify the simple bounded weight modules of ${mathfrak{sl}(infty})$, ${mathfrak{o}(infty)}$ and ${mathfrak{sp}(infty)}$, and compute their annihilators in $U({mathfrak{sl}(infty}))$, $U({mathfrak{o}(infty))}$, $U({mathfrak{sp}(infty))}$, respectively.
We prove that the negative infinitesimal generator $L$ of a semigroup of positive contractions on $L^infty$ has a bounded $H^infty(S_eta^0)$-calculus on the associated Poisson semigroup-BMO space for any angle $eta>pi/2$, provided the semigroup satisfies Bakry-Emrys $Gamma_2 $ criterion. Our arguments only rely on the properties of the underlying semigroup and works well in the noncommutative setting. A key ingredient of our argument is a quasi monotone property for the subordinated semigroup $T_{t,alpha}=e^{-tL^alpha},0<alpha<1$, that is proved in the first half of the article.
Let $1leq p,q < infty$ and $1leq r leq infty$. We show that the direct sum of mixed norm Hardy spaces $big(sum_n H^p_n(H^q_n)big)_r$ and the sum of their dual spaces $big(sum_n H^p_n(H^q_n)^*big)_r$ are both primary. We do so by using Bourgains localization method and solving the finite dimensional factorization problem. In particular, we obtain that the spaces $big(sum_{nin mathbb N} H_n^1(H_n^s)big)_r$, $big(sum_{nin mathbb N} H_n^s(H_n^1)big)_r$, as well as $big(sum_{nin mathbb N} BMO_n(H_n^s)big)_r$ and $big(sum_{nin mathbb N} H^s_n(BMO_n)big)_r$, $1 < s < infty$, $1leq r leq infty$, are all primary.
$SL^infty$ denotes the space of functions whose square function is in $L^infty$, and the subspaces $SL^infty_n$, $ninmathbb{N}$, are the finite dimensional building blocks of $SL^infty$. We show that the identity operator $I_{SL^infty_n}$ on $SL^infty_n$ well factors through operators $T : SL^infty_Nto SL^infty_N$ having large diagonal with respect to the standard Haar system. Moreover, we prove that $I_{SL^infty_n}$ well factors either through any given operator $T : SL^infty_Nto SL^infty_N$, or through $I_{SL^infty_N}-T$. Let $X^{(r)}$ denote the direct sum $bigl(sum_{ninmathbb{N}_0} SL^infty_nbigr)_r$, where $1leq r leq infty$. Using Bourgains localization method, we obtain from the finite dimensional factorization result that for each $1leq rleq infty$, the identity operator $I_{X^{(r)}}$ on $X^{(r)}$ factors either through any given operator $T : X^{(r)}to X^{(r)}$, or through $I_{X^{(r)}} - T$. Consequently, the spaces $bigl(sum_{ninmathbb{N}_0} SL^infty_nbigr)_r$, $1leq rleq infty$, are all primary.
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