We review the basic facts about the theory of paracommutators in Rn (sec S. Janson, J. Peetre, Trans. Am. Math. Soc. 305 (1988), 467504). We also give an interpretation of paracommutators from the point of view of group representations. This suggests a generalization to more general groups. Here we sketch a theory of paracommutators over stratified groups. This include the famous Heisenberg group. Finally, we take up the question of generalizing the notion of Schatten-von Neumann trace ideals to...

Let E be a Banach space. We consider a Cauchy problem of the type ⎧ $D_t^{k}u+{\sum }_{j=0}^{k-1}{\sum }_{\left|\alpha \right|\le m}{A}_{j,\alpha }\left(D_t^j{D}_x^{\alpha }u\right)=f$ in ${ℝ}^{n+1}$, ⎨ ⎩ $D_t^{j}u(0,x)={\phi }_j\left(x\right)$ in ${ℝ}^n$, j=0,...,k-1, where each $A_{j,\alpha }$ is a given continuous linear operator from E into itself. We prove that if the operators $A_{j,\alpha }$ are nilpotent and pairwise commuting, then the problem is well-posed in the space of all functions $u\in C^{\infty }({ℝ}^{n+1},E)$ whose derivatives are equi-bounded on each bounded subset of ${ℝ}^{n+1}$.

Beurling's classical theorem gives a complete characterization of all invariant subspaces in the Hardy space H²(D). To generalize the theorem to higher dimensions, one is naturally led to determining the structure of each unitary equivalence (resp. similarity) class. This, in turn, requires finding podal (resp. s-podal) points in unitary (resp. similarity) orbits. In this note, we find that H-outer (resp. G-outer) functions play an important role in finding podal (resp. s-podal) points. By the methods...

We prove that for each linear contraction T : X → X (∥T∥ ≤ 1), the subspace F = {x ∈ X : Tx = x} of fixed points is 1-complemented, where X is a suitable subspace of L¹(E*) and E* is a separable dual space such that the weak and weak* topologies coincide on the unit sphere. We also prove some related fixed point results.

We distinguish a class of unbounded operators in ${}^r$, r ≥ 1, related to the self-adjoint operators in ². For these operators we prove a kind of individual ergodic theorem, replacing the classical Cesàro averages by Borel summability. The result is equivalent to a version of Gaposhkin’s criterion for the a.e. convergence of operators. In the proof, the theory of martingales and interpolation in ${}^r$-spaces are applied.

For a wide class of weights we find the approximative point spectrum and the essential spectrum of the pointwise multiplication operator $M_{\phi }$, $M_{\phi }\left(f\right)=\phi f$, on the weighted Banach spaces of analytic functions on the disc with the sup-norm. Thus we characterize when $M_{\phi }^{\text{'}}$ is Fredholm or is an into isomorphism. We also study cyclic phenomena for the adjoint map $M_{\phi }^{\text{'}}$.

We introduce generalized Campanato spaces ${}_{p,\varphi }$ on a probability space (Ω,ℱ,P), where p ∈ [1,∞) and ϕ: (0,1] → (0,∞). If p = 1 and ϕ ≡ 1, then ${}_{p,\varphi }=BMO$. We give a characterization of the set of all pointwise multipliers on ${}_{p,\varphi }$.

A bounded linear operator T defined on a Banach space X is said to be supercyclic if there exists a vector x ∈ X such that the projective orbit {λTⁿx : λ ∈ ℂ, n ∈ ℕ} is dense in X. The aim of this survey is to show the relationship between positivity and supercyclicity. This relationship comes from the so called Positive Supercyclicity Theorem. Throughout this exposition, interesting new directions and open problems will appear.

We first determine when the sum of products of Hankel and Toeplitz operators is equal to zero; then we characterize when the product of a Toeplitz operator and a Hankel operator is a compact perturbation of a Hankel operator or a Toeplitz operator and when it is a finite rank perturbation of a Toeplitz operator.

The investigation of properties of generalized Toeplitz operators with respect to the pairs of doubly commuting contractions (the abstract analogue of classical two variable Toeplitz operators) is proceeded. We especially concentrate on the condition of existence such a non-zero operator. There are also presented conditions of analyticity of such an operator.