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Register a new userWeakly Separated Bessel Systems of Model Spaces
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Authors
Alberto Dayan
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We show that any weakly separated Bessel system of model spaces in the Hardy space on the unit disc is a Riesz system and we highlight some applications to interpolating sequences of matrices. This will be done without using the recent solution of the Feichtinger conjecture, whose natural generalization to multi-dimensional model sub-spaces of $mathrm{H}^2$ turns out to be false.
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Let $mathcal{D}$ be the class of radial weights on the unit disk which satisfy both forward and reverse doubling conditions. Let $g$ be an analytic function on the unit disk $mathbb{D}$. We characterize bounded and compact Volterra type integration operators [ J_{g}(f)(z)=int_{0}^{z}f(lambda)g(lambda)dlambda ] between weighted Bergman spaces $L_{a}^{p}(omega )$ induced by $mathcal{D}$ weights and Hardy spaces $H^{q}$ for $0<p,q<infty$.
In this article we examine Dirichlet type spaces in the unit polydisc, and multipliers between these spaces. These results extend the corresponding work of G. D. Taylor in the unit disc. In addition, we consider functions on the polydisc whose restrictions to lower dimensional polydiscs lie in the corresponding Dirichet type spaces. We see that such functions need not be in the Dirichlet type space of the whole polydisc. Similar observations are made regarding multipliers.
We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form $k(s,u) = sum a_n n^{-s-bar u}$, and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing kernel Hilbert spaces to be the same, and introduce a notion of weak isomorphism. Many of the spaces we consider turn out to be weakly isomorphic as reproducing kernel Hilbert spaces to the Drury-Arveson space $H^2_d$ in $d$ variables, where $d$ can be any number in ${1,2,ldots, infty}$, and in particular their multiplier algebras are unitarily equivalent to the multiplier algebra of $H^2_d$. Thus, a family of multiplier algebras of Dirichlet series are exhibited with the property that every complete Pick algebra is a quotient of each member of this family. Finally, we determine precisely when such a space of Dirichlet series is weakly isomorphic to $H^2_d$ and when its multiplier algebra is isometrically isomorphic to $Mult(H^2_d)$.
For $-1<alpha<infty$, let $omega_alpha(z)=(1+alpha)(1-|z|^2)^alpha$ be the standard weight on the unit disk. In this note, we provide descriptions of the boundedness and compactness for the Toeplitz operators $T_{mu,beta}$ between distinct weighted Bergman spaces $L_{a}^{p}(omega_{alpha})$ and $L_{a}^{q}(omega_{beta})$ when $0<pleq1$, $q=1$, $-1<alpha,beta<infty$ and $0<pleq 1<q<infty, -1<betaleqalpha<infty$, respectively. Our results can be viewed as extensions of Pau and Zhaos work in cite{Pau}. Moreover, partial of main results are new even in the unweighted settings.
It is known in Hilbert space frame theory that a Bessel sequence can be expanded to a frame. Contrary to Hilbert space situation, using a result of Casazza and Christensen, we show that there are Banach spaces and approximate Bessel sequences which can not be expanded to approximate Schauder frames. We characterize Banach spaces in which one can expand approximate Bessel sequences to approximate Schauder frames.
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