We present a Banach space $X_\alpha$, equipped with an explicit norm, whose properties resemble those of the predual $JT_*$ of the James Tree space. In particular, $X_\alpha$ admits a shrinking dyadic basis with the property that the sums of its elements over any segment are uniformly bounded. The space $X_\alpha$ is complementably $\ell_2$-saturated, and its dual $X_\alpha^*$ does not contain an isomorphic copy of $\ell_1$. Finally, the bidual $X_\alpha^{**}$ is also complementably $\ell_2$-saturated, and the quotient $X_\alpha^{**}/X_\alpha$ is isometric to $c_0(2^{\mathbb{N}})$.
We prove that the Hilbert matrix operator $H:A^2_{\ln^{\alpha}} \to A^2$ is bounded when $\alpha\geq 2$ and that $H$ does not map $A^2_{\ln^{\alpha}} $ into $A^2$ when $\alpha<2$, solving an open problem posed by B. Karapetrovi\'c and M. Jevti\'c. The boundedness when $\alpha=2$ is shown as a corollary to the more gene...
We construct a one-parameter family $(X_c)_{0<c<1}$ of real reflexive Banach spaces, isomorphic to $\ell_2$ and satisfying $d_{\mathrm{BM}}(X_c,\ell_2)\le c^{-1}$, such that $(X_c,X_c)$ fails the weak maximizing property while $(X_c^*,X_c^*)$ has it. The failure of the WMP for $X_c$ is witnessed by a diagonal operator...
Domingo García, M. Maestre, D. Pinasco et al.· 0 citations
Let $X$ be a normed space and $S_X$ be the unit sphere of $X$. We say that $X$ has Property (H) if there exist two increasing sequences $V_j\subset X$ and $H_j\subset \ell_2$, $j=1,2,\ldots$, of finite dimensional subspaces such that $V=\bigcup_j V_j$ is dense in $X$, and there is a uniformly continuous mapping $\psi:S...
We study the reciprocal problem for weighted Bergman spaces: if $f\in A_\alpha^p(\Bn)$ and $\inf_{\Bn}|f|>0$, does it follow that $1/f\in A_\alpha^p(\Bn)$? We determine the range of parameters for which the answer is affirmative. In particular, we prove that functions in $A_{\alpha}^p(\D)$ have the reciprocal property...
Let $\omega: \mathbb{N} \to [1, \infty)$ be a weight on the multiplicative semigroup of natural numbers. We investigate the Bochner-Schoenberg-Eberlein (BSE) property for the weighted Banach algebra of Dirichlet series $A_\omega = \ell^1(\mathbb{N}, \omega)$ under Dirichlet convolution. By employing Bohr's corresponden...
A Banach space $X$ is $L$-embedded in its bidual if there is a projection $P\colon X^{**}\to X$ such that $\|x\|=\|Px\|+\|x-Px\|$ for all $x\in X^{**}$. We show that, as long as $X$ has dimension at least $2$, its Lipschitz-free space, denoted by $\mathcal{F}(X)$, is not $L$-embedded in its bidual. Our methods have app...
Bruno de Mendonça Braga, C. Gartland, G. Lancien et al.· 2 citations· ⚡1
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