We examine the generalized Hilbert (matrix) operators $$ H_{g,\gamma} : (a_n) \mapsto \sum_{n=1}^{\infty} \bigg(\frac{k}{n}\bigg)^{\gamma} \frac{g_k a_n}{n+k} $$ on the $\ell^p$ spaces, $1<p<\infty$, where $g=(g_n)$ is a sequence and $-1/p<\gamma<1-1/p$. Given a partition of the natural numbers $\bigcup_j I_j = \mathbb N$, we encode $g$ to our investigation via its $\ell^p$-means $G_j$ over the sets $I_j$. We prove that if $I_j$ increases exponentially, the data $(G_j)$ characterizes boundedness, but is insufficient to determine the exact value of the essential norm. Nonetheless, if the growth rate of $(I_j)$ is subexponential, the corresponding data $(G_j)$, given that the tail converge, is sufficient to determine the exact value, in which case we also calculate it. Moreover, it is shown that the boundedness of $(G_j)$ is sufficient, but not necessary (and necessary, but not sufficient) to ensure $H_{g,\gamma}$ is bounded if the underlying partition $(I_j)$ increases subexponentially (and superexponentially, respectively). Using the dual operator, we also prove that the essential norm of $H_{g,\gamma}$ is comparable with $\limsup_j G(j)$ when $(I_j)$ grows exponentially.
We compute the large-$N$ asymptotics of the determinant of the identity plus or minus a generalized Hilbert matrix. For $N \in \mathbb{N}$ and $\delta \in \mathbb{R} \setminus \{-1, -2, \ldots\}$, let \[ H_N^\delta := \left( \frac{\sin((1+\delta)\pi)}{\pi(j+k+1+\delta)} \right)_{j,k=0}^{N-1} \] be the generalized Hilbe...
Let $g$ be analytic in the unit disc and consider the generalized Hilbert operator $$ \mathcal{H}_g(f)(z)=\int_0^1 f(t)g'(tz)\, dt. $$ The boundedness of $\mathcal H_g$ on $H^p$ is characterized by the mean Lipschitz condition $g\in\Lambda\left(p,\frac{1}{p}\right)$ when $1<p\leq2$, while the problem remains open for $...
D. Norrbo, J. A. Pel'aez, Fang-Lei Wu· 2 citations· ⚡1
Let $P_n$ be the matrix of a random permutation of $n$ symbols and let $M_n=\log\max_{|z|=1}|\det(I-zP_n)|$. Cook and Zeitouni proved that $M_n/\log n$ converges in probability to a constant $x_0$ for a uniform permutation. We show that the $\sqrt{\log n}$ fluctuations of $M_n$ are carried entirely by the number of cyc...
Let $\mu$ be a finite positive Borel measure on $[0,1)$ and let $\gamma>0$. We establish sharp mapping criteria for the generalized Ces\`aro operator \begin{equation*} \mathcal C_{\mu,\gamma}f(z) =\sum_{n=0}^\infty \mu_n \left(\sum_{k=0}^n \frac{\Gamma(n-k+\gamma)}{\Gamma(\gamma)(n-k)!}a_k\right)z^n,\qquad z\in \mathbb...
Let $W=W(B_n)$ act diagonally on $\mathfrak{h}\oplus\mathfrak{h}^*$, let $S=\mathbb{C}[\mathfrak{h}\oplus\mathfrak{h}^*]$, let $J\subset S$ be the ideal generated by the $W$-alternating polynomials and $\mathfrak{m}_S$ is the maximal ideal of the origin. For sufficiently large $m$ we compute $q,t$-Fuss-Catalan polynomi...
We call an $n$-tuple of $2 \times 2$ matrices a ``\textit{$2 \times 2$ matrix $\Gamma_n$-contraction}''if its joint eigenvalues (joint spectrum for a commuting tuple of matrices) are contained in $\Gamma_n$. In this short note on $\Gamma_n$-contraction, we produce two families of $2 \times 2$ matrix $\Gamma_n$-contract...
Bhaskar Paul· 0 citations
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