These lectures will focus on those properties of maximal monotone operators which are valid in arbitrary real Banach spaces.

On définit et étudie les espaces de Banach de type et de cotype $p$, $0\<p\le 2$. Cela permet de dire que, dès que certaines applications sont $q$-sommantes, elles sont aussi $r$-sommantes pour $r\le q$.

In many fields: signal theory, economics, etc... where random functions are introduced, frequently and naturally we encounter integral operators whose kernels are covariances. Of course the most immediate properties of that particular kind of integral operators have already been published, this allows us to quote them without proofs. But these properties are scattered over, so we have thought useful to present here a synthetic ordered, almost full account of them without pretending to originality,...

Nous prouvons l’hyper-réflexivité du shift bilatéral $S_{\omega }$ sur ${\ell }_{\omega }^2\left(\mathbb{Z}\right)$, lorsque le poids vérifie $\omega \left(n\right)=1$ for $n\ge 0$ et ${lim}_{n\to -\infty }\omega \left(n\right)=+\infty$.

In this paper we show that from an estimate of the form $su{p}_{t\ge 0}\left|\right|C\left(t\right)-cos\left(at\right)I\left|\right|<1$, we can conclude that C(t) equals cos(at)I. Here ${\left(C\left(t\right)\right)}_{t\ge 0}$ is a strongly continuous cosine family on a Banach space.

A homology theory of Banach manifolds of a special form, called FSQL-manifolds, is developed, and also a homological degree of FSQL-mappings between FSQL-manifolds is introduced.

We show that if $\alpha >1$, then the logarithmically weighted Bergman space $A_{{log}^{\alpha }}^2$ is mapped by the Libera operator $ℒ$ into the space $A_{{log}^{\alpha -1}}^2$, while if $\alpha >2$ and $0<\epsilon \le \alpha -2$, then the Hilbert matrix operator $H$ maps $A_{{log}^{\alpha }}^2$ into $A_{{log}^{\alpha -2-\epsilon }}^2$.We show that the Libera operator $ℒ$ maps the logarithmically weighted Bloch space ${ℬ}_{{log}^{\alpha }}$, $\alpha \in ℝ$, into itself, while $H$ maps ${ℬ}_{{log}^{\alpha }}$ into ${ℬ}_{{log}^{\alpha +1}}$.In Pavlović’s paper (2016) it is shown that $ℒ$ maps the logarithmically weighted Hardy-Bloch space ${ℬ}_{{log}^{\alpha }}^1$, $\alpha >0$, into ${ℬ}_{{log}^{\alpha -1}}^1$. We show that this result is sharp. We also show that $H$ maps ${ℬ}_{{log}^{\alpha }}^1$, $\alpha \ge 0$, into ${ℬ}_{{log}^{\alpha -1}}^1$ and...

It is proved that if $J_i$ is a Jordan operator on a Hilbert space with the Jordan decomposition $J_i=N_i+Q_i$, where $N_i$ is normal and $Q_i$ is compact and quasinilpotent, i = 1,2, and the Lie algebra generated by J₁,J₂ is an Engel Lie algebra, then the Banach algebra generated by J₁,J₂ is an Engel algebra. Some results for normal operators and Jordan operators on Banach spaces are given.

Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie groups modelled on locally convex spaces. In the present paper, we generalise this construction and show that differentiable mappings on a compact manifold (possibly with boundary) with values in a Lie groupoid form infinite-dimensional Lie groupoids which we...