Operator Theory and Its Applications: In Memory of V. B. Lidskii (1924-2008)

This book is a collection of articles devoted to the theory of linear operators in Hilbert spaces and its applications. The subjects covered range from the abstract theory of Toeplitz operators to the analysis of very specific differential operators arising in quantum mechanics, electromagnetism, and the theory of elasticity; the stability of numerical methods is also discussed. Many of the articles deal with spectral problems for not necessarily selfadjoint operators. Some of the articles are surveys outlining the current state of the subject and presenting open problems.

On a class of nonselfadjoint periodic boundary value problems with discrete real spectrum

In a previous paper (J. of Differential Equations, Vol. 249 (2010), 3081-3098) we examined a family of periodic Sturm-Liouville problems with boundary and interior singularities which are highly non-self-adjoint but have only real eigenvalues. We now establish Schatten class properties of the associated resolvent operator.

On the near periodicity of eigenvalues of Toeplitz matrices

Let $A$ be an infinite Toeplitz matrix with a real symbol $f$ defined on $[-\pi, \pi]$. It is well known that the sequence of spectra of finite truncations $A_N$ of $A$ converges to the convex hull of the range of $f$. Recently, Levitin and Shargorodsky, on the basis of some numerical experiments, conjectured, for symbols $f$ with two discontinuities located at rational multiples of $\pi$, that the eigenvalues of $A_N$ located in the gap of $f$ asymptotically exhibit periodicity in $N$, and suggested a formula for the period as a function of the position of discontinuities. In this paper, we quantify and prove the analog of this conjecture for the matrix $A^2$ in a particular case when $f$ is a piecewise constant function taking values $-1$ and $1$.

La Faune marine dans le voyage de Saint brendan anglo-normand

Le Voyage de Saint Brendan présente un univers hybride, à la fois suffisamment réaliste pour y reconnaître des phénomènes scientifiquement vérifiables et profondément symbolique dans ses connotations. Les animaux marins qui y figurent ne sont en rien anthropomorphes, mais se démarquent aussi nettement de l'image donnée par les bestiaires, dans la mesure où leurs actions et caractéristiques ne sont pas en soi porteuses de sens. La faune aquatique est à la fois une menace et une présence protectrice, la providence divine déterminant de cas en cas la nature des rencontres entre moines et poissons, baleines et monstres marins. Ces bêtes sont en elles-mêmes moralement neutres, guidées par l'instinct; mais Benedeit exploite ce trait pour mettre en exergue la soumission de ces créatures à la volonté du Créateur, et leur importance pour Brendan et ses compagnons.

'Nature concocts & expels': the agents and processes of recovery from disease in Early Modern England

The ‘golden saying’ in early modern medicine was ‘Nature is the healer of disease’. This article uncovers the meaning and significance of this forgotten axiom by investigating perceptions of the agents and physiological processes of recovery from illness in England, c.1580-1720. Drawing on sources such as medical texts and diaries, it shows that doctors and laypeople attributed recovery to three agents – God, Nature, and the practitioner. While scholars are familiar with the roles of providence and medicine, the vital agency of Nature has been overlooked. In theory, the agents operated in a hierarchy: Nature was ‘God’s instrument’, and the physician, ‘Nature’s servant’; but in practice the power balance was more ambivalent. Nature was depicted both as a housewife who cooked and cleaned the humours, and as a warrior, who defeated the disease. Through exploring these complex dynamics, the article sheds fresh light on concepts of gender, disease, and bodies.