The multiplicative Hilbert matrix

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• 作者：Ole Fredrik Brevig^a ; ^{ole.brevig@math.ntnu.no" class="auth_mail" title="E-mail the corresponding author} ; Karl-Mikael Perfekt^a ; ^{karl-mikael.perfekt@math.ntnu.no" class="auth_mail" title="E-mail the corresponding author} ; Kristian Seip^a ; ^{seip@math.ntnu.no" class="auth_mail" title="E-mail the corresponding author} ; Aristomenis G. Siskakis^b ; ^{siskakis@math.auth.gr" class="auth_mail" title="E-mail the corresponding author} ; Dragan Vukotić^c ; ^{dragan.vukotic@uam.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11M99 ; 42B30 ; 47B35 ; 47G10
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：22 October 2016
• 年：2016
• 卷：302
• 期：Complete
• 页码：410-432
• 全文大小：460 K

文摘

It is observed that the infinite matrix with entries ${(\sqrt{mn}\log (mn))}^{-1}$ for b34e4b9636639" title="Click to view the MathML source">m,n≥2$m,n\ge 2$ appears as the matrix of the integral operator $\mathbf{H}f(s):={\int }_{1/2}^{+\infty }f(w)(\zeta (w+s)-1)dw$ with respect to the basis (n^−s)_n≥2${(n^{-s})}_{n\ge 2}$; here ζ(s)$\zeta (s)$ is the Riemann zeta function and H is defined on the Hilbert space ${\mathcal{H}}_0^2$ of Dirichlet series vanishing at +∞ and with square-summable coefficients. This infinite matrix defines a multiplicative Hankel operator according to Helson's terminology or, alternatively, it can be viewed as a bona fide (small) Hankel operator on the infinite-dimensional torus T^∞${\mathbb{T}}^{\infty }$. By analogy with the standard integral representation of the classical Hilbert matrix, this matrix is referred to as the multiplicative Hilbert matrix. It is shown that its norm equals π   and that it has a purely continuous spectrum which is the interval [0,π]$[0,\pi ]$; these results are in agreement with known facts about the classical Hilbert matrix. It is shown that the matrix (m^1/pn^(p−1)/plog⁡(mn))⁻¹${(m^{1/p}{n}^{(p-1)/p}\log (mn))}^{-1}$ has norm π/sin⁡(π/p)$\pi /\sin (\pi /p)$ when acting on ℓ^p${\ell }^p$ for 1<p<∞$1<p<\infty$. However, the multiplicative Hilbert matrix fails to define a bounded operator on ${\mathcal{H}}_0^p$ for p≠2$p\ne 2$, where ${\mathcal{H}}_0^p$ are H^p$H^p$ spaces of Dirichlet series. It remains an interesting problem to decide whether the analytic symbol ∑_n≥2(log⁡n)⁻¹n^−s−1/2${\sum }_{n\ge 2}{(\log n)}^{-1}{n}^{-s-1/2}$ of the multiplicative Hilbert matrix arises as the Riesz projection of a bounded function on the infinite-dimensional torus T^∞${\mathbb{T}}^{\infty }$.