A Helson matrix with explicit eigenvalue asymptotics

A Helson matrix (also known as a multiplicative Hankel matrix) is an infinite matrix with entries $\{a(jk)\}$ for $j,k\geq1$. Here the $(j,k)$'th term depends on the product $jk$. We study a self-adjoint Helson matrix for a particular sequence $a(j)=(\sqrt{j}\log j(\log\log j)^\alpha))^{-1}$, $j\geq 3$, where $\alpha>0$, and prove that it is compact and that its eigenvalues obey the asymptotics $\lambda_n\sim\varkappa(\alpha)/n^\alpha$ as $n\to\infty$, with an explicit constant $\varkappa(\alpha)$. We also establish some intermediate results (of an independent interest) which give a connection between the spectral properties of a Helson matrix and those of its continuous analogue, which we call the integral Helson operator

Projecting onto Helson matrices in Schatten classes

A Helson matrix is an infinite matrix $A = (a_{m,n})_{m,n\geq1}$ such that the entry $a_{m,n}$ depends only on the product $mn$. We demonstrate that the orthogonal projection from the Hilbert--Schmidt class $\mathcal{S}_2$ onto the subspace of Hilbert--Schmidt Helson matrices does not extend to a bounded operator on the Schatten class $\mathcal{S}_q$ for $1 \leq q \neq 2 < \infty$. In fact, we prove a more general result showing that a large class of natural projections onto Helson matrices are unbounded in the $\mathcal{S}_q$-norm for $1 \leq q \neq 2 < \infty$. Two additional results are also presented.Comment: This paper has been has been accepted for publication in Studia Mat

Convolution operators and function algebras

this thesis we investigate questions on convolution operators and function algebras, as well as generalising a result from from function algebras to the noncommutuative setting. After detailing the necessary background in Chapter 1, Chapter 2 deals with convolution operators on Banach lattices. \~le start by describing certain Banach lattices of functions defined on a locally compact abelian group G which generalises a class of Banach lattices introduced in Johansson (2008).EThOS - Electronic Theses Online ServiceGBUnited Kingdo

RANTES release by human adipose tissue in vivo and evidence for depot-specific differences

Obesity is associated with elevated inflammatory signals from various adipose tissue depots. This study aimed to evaluate release of regulated on activation, normal T cell expressed and secreted (RANTES) by human adipose tissue in vivo and ex vivo, in reference to monocyte chemoattractant protein-1 (MCP-1) and interleukin-6 (IL-6) release. Arteriovenous differences of RANTES, MCP-1, and IL-6 were studied in vivo across the abdominal subcutaneous adipose tissue in healthy Caucasian subjects with a wide range of adiposity. Systemic levels and ex vivo RANTES release were studied in abdominal subcutaneous, gastric fat pad, and omental adipose tissue from morbidly obese bariatric surgery patients and in thoracic subcutaneous and epicardial adipose tissue from cardiac surgery patients without coronary artery disease. Arteriovenous studies confirmed in vivo RANTES and IL-6 release in adipose tissue of lean and obese subjects and release of MCP-1 in obesity. However, in vivo release of MCP-1 and RANTES, but not IL-6, was lower than circulating levels. Ex vivo release of RANTES was greater from the gastric fat pad compared with omental (P = 0.01) and subcutaneous (P = 0.001) tissue. Epicardial adipose tissue released less RANTES than thoracic subcutaneous adipose tissue in lean (P = 0.04) but not obese subjects. Indexes of obesity correlated with epicardial RANTES but not with systemic RANTES or its release from other depots. In conclusion, RANTES is released by human subcutaneous adipose tissue in vivo and in varying amounts by other depots ex vivo. While it appears unlikely that the adipose organ contributes significantly to circulating levels, local implications of this chemokine deserve further investigation