arXiv subjects
Publications and source records attributed to Vassilis Daskalogiannis.
This article aims to explore the most recent developments in the study of the Hilbert matrix, acting as an operator on spaces of analytic functions and sequence spaces. We present the latest advances in this area, aiming to provide a concise overview for researchers interested in delving into the captivating theory of operator matrices.
In this article we study the generalized Hilbert matrix operator $Γ_μ$ acting on the Bergman spaces $A^p$ of the unit disc for $1\leq p<\infty$. In particular, we characterize the measures $μ$ for which the operator $Γ_μ$ is bounded and we provide estimates of its operator norm. Finally, we also describe when $Γ_μ$ is compact by computing its essential norm.
For a finite, positive, Borel measure $μ$ on $(0,1)$ we consider an infinite matrix $Γ_μ$, related to the classical Hausdorff matrix defined by the same measure $μ$, in the same algebraic way that the Hilbert matrix is related to the Cesáro matrix. When $μ$ is the Lebesgue measure, $Γ_μ$ reduces to the classical Hilbert matrix. We prove that the matrices $Γ_μ$ are not Hankel, unless $μ$ is a constant multiple of the Lebesgue measure, we give necessary and sufficient conditions for their boundedness on the scale of Hardy spaces $H^p, \, 1 \leq p < \infty$, and we study their compactness and complete continuity properties. In the case $2\leq p<\infty$, we are able to compute the exact value of the norm of the operator.
We prove the nontrivial variant \[ \sum\limits_{m,n=1}^{\infty}\Big(\frac{n}{m}\Big)^{\frac{1}{q}-\frac{1}{p}}\frac{a_mb_n}{m+n-1}\leq\fracπ{\sin\fracπ{p}} \Big( \sum\limits_{m=1}^{\infty}a_m^p\Big)^{\frac 1p}\Big( \sum\limits_{n=1}^{\infty}b_n^q\Big)^{\frac 1q} \] of the well known Hilbert's inequality. Then we use this to determine the exact value $\fracπ{\sin\fracπp}$ of the norm of the Hilbert matrix as an operator acting on the Hardy-Littlewood space $K^p$. This space consists of all functions $f(z)=\sum\limits_{m=0}^{\infty}a_mz^m$ analytic in the unit disc with $\|f\|_{K^p}^p=\sum\limits_{m=0}^{\infty}(m+1)^{p-2}|a_m|^p<+\infty$.
It is a classical theorem of Sarason that an analytic function of bounded mean oscillation ($BMOA$), is of vanishing mean oscillation if and only if its rotations converge in norm to the original function as the angle of the rotation tends to zero. In a series of two papers Blasco et al. have raised the problem of characterizing all semigroups of holomorphic functions $(φ_t)$ that can replace the semigroup of rotations in Sarason's Theorem. We give a complete answer to this question, in terms of a logarithmic vanishing oscillation condition on the infinitesimal generator of the semigroup $(φ_t)$. In addition we confirm the conjecture of Blasco et al. that all such semigroups are elliptic. We also investigate the analogous question for the Bloch and the little Bloch space and surprisingly enough we find that the semigroups for which the Bloch version of Sarason's Theorem holds are exactly the same as in the $BMOA$ case.
The aim of this article is to study semigroups of composition operators on the BMOA-type spaces $BMOA_p$, and on their "little oh" analogues $VMOA_p$. The spaces $BMOA_p$ were introduced by R. Zhao as part of the large family of F(p,q,s) spaces, and are the Möbius invariant subspaces of the Dirichlet spaces $D^p_{p-1}$. We study the maximal subspace of strong continuity, providing a sufficient condition on the infinitesimal generator of $ϕ$, under which $[ϕ_t,BMOA_p]=VMOA_p$, and a related necessary condition in the case where the Denjoy - Wolff point of the semigroup is in $\mathbb{D}$. Further, we characterize those semigroups, for which $[ϕ_t, BMOA_p]=VMOA_p$, in terms of the resolvent operator of the infinitesimal generator of $T_t$. In addition we provide a connection between the maximal subspace of strong continuity and the Volterra-type operators $T_g$. We characterize the symbols g for which $T_g$ acting from $BMOA$ to $BMOA_1$ is bounded or compact, thus extending a related result to the case $p=1$. We also prove that for $1<p<2$ compactness of $T_g$ on $BMOA_p$ is equivalent to weak compactness.