Our concern is with the group of conformal transformations of a finite-dimensional real quadratic space of signature (p,q), that is one that is isomorphic to ${ℝ}^{p,q}$, the real vector space ${ℝ}^{p+q}$, furnished with the quadratic form $x^{\left(2\right)}=x·x=-x_1^2-x_2^2-...-x_p^2+x_{p+1}^2+...+x_{p+q}^2$, and especially with a description of this group that involves Clifford algebras.

Soient ${𝒦}_1$ et ${𝒦}_2$ deux espaces de Krein de fonctions analytiques dans le disque unité invariants pour l’opérateur de déplacement à gauche $R_0(R_{0}f\left(z\right)=\left(f\right(z)-f(0\left)\right)/z)$ et soit $A$ un opérateur linéaire continu de ${𝒦}_1$ dans ${𝒦}_2$ dont l’adjoint commute avec $R_0$. Nous étudions les dilatations $B$ de $A$ qui conservent cette propriété de commutation et pour lesquelles les formes hermitiennes définies par $I-A{A}^{*}$ et $I-B{B}^{*}$ ont le même nombre de carrés négatifs. Nous obtenons ainsi une version du théorème de dilatation des commutants d’opérateurs dans le cadre...

Integral equations of the form (2) below, dual to (1) are studied from the point of view of finding their effective solutions, the results being given in Section 1. The results are applied in Section 2 for solving nonlocal problems for the polyharmonic functions in the half plane.

The notion of a $J^3$-triple is studied in connection with a geometrical approach to the generalized Hurwitz problem for quadratic or bilinear forms. Some properties are obtained, generalizing those derived earlier by the present authors for the Hurwitz maps S × V → V. In particular, the dependence of each scalar product involved on the symmetry or antisymmetry is discussed as well as the configurations depending on various choices of the metric tensors of scalar products of the basis elements. Then...

The authors are dealing with the Dirichlet integral-type biholomorphic-invariant pseudodistance ${\rho }_X^{\alpha }(z_0,z)\left[\right]$ introduced by Dolbeault and Ławrynowicz (1989) in connection with bordered holomorphic chains of dimension one. Several properties of the related hyperbolic-like manifolds are considered remarking the analogies with and differences from the familiar hyperbolic and Stein manifolds. Likewise several examples are treated in detail.

The point equation of the associated curve of the indefinite numerical range is derived, following Fiedler’s approach for definite inner product spaces. The classification of the associated curve is presented in the $3\times 3$ indefinite case, using Newton’s classification of cubic curves. Illustrative examples of all the different possibilities are given. The results obtained extend to Krein spaces results of Kippenhahn on the classical numerical range.

One of the fundamental objectives of the theory of symplectic singularities is to study the symplectic invariants appearing in various geometrical contexts. In the paper we generalize the symplectic cohomological invariant to the class of generalized canonical mappings. We analyze the global structure of Lagrangian Grassmannian in the product symplectic space and describe the local properties of generic symplectic relations.

In this paper we characterize Moore-Penrose inverses of Gram matrices leaving a cone invariant in an indefinite inner product space using the indefinite matrix multiplication. This characterization includes the acuteness (or obtuseness) of certain closed convex cones.

Contents Introduction 119 1. Quasiregular mappings 120 2. The Beltrami equation 121 3. The Beltrami-Dirac equation 122 4. A quest for compactness 124 5. Sharp $L^p$-estimates versus variational integrals 125 6. Very weak solutions 128 7. Nonlinear commutators 129 8. Jacobians and wedge products 131 9. Degree formulas 134 References 136

Consider a family of integral operators and a related family of differential operators, both defined on a class of analytic functions holomorphic in the unit disk, distortion properties of the real part are derived from a general aspect.

In this paper the hyponormal operators on Krein spaces are introduced. We state conditions for the hyponormality of bounded operators focusing, in particular, on those operators $T$ for which there exists a fundamental decomposition $𝕂={𝕂}^{+}\oplus {𝕂}^{-}$ of the Krein space $𝕂$ with ${𝕂}^{+}$ and ${𝕂}^{-}$ invariant under $T$.