Operator pencil passing through a given operator

Let $\Delta$ be a linear differential operator acting on the space of densities of a given weight \lo on a manifold $M$. One can consider a pencil of operators \hPi(\Delta)=\{\Delta_\l\} passing through the operator $\Delta$ such that any \Delta_\l is a linear differential operator acting on densities of weight \l. This pencil can be identified with a linear differential operator \hD acting on the algebra of densities of all weights. The existence of an invariant scalar product in the algebra of densities implies a natural decomposition of operators, i.e. pencils of self-adjoint and anti-self-adjoint operators. We study lifting maps that are on one hand equivariant with respect to divergenceless vector fields, and, on the other hand, with values in self-adjoint or anti-self-adjoint operators. In particular we analyze the relation between these two concepts, and apply it to the study of \diff(M)-equivariant liftings. Finally we briefly consider the case of liftings equivariant with respect to the algebra of projective transformations and describe all regular self-adjoint and anti-self-adjoint liftings.Comment: 32 pages, LaTeX fil

Fourier analysis of a space of Hilbert-Schmidt operators-new Ha-plitz type operators

If a group acts via unitary operators on a Hilbert space of functions then this group action extends in an obvious way to the space of Hilbert- Schmidt operators over the given Hilbert space. Even if the action on functions is irreducible, the action on H.S . operators need not be irreducible. It is often of considerable interest to find out what the irreducible constituents are. Such an attitude has recently been advocated in the theory of "Ha-plitz" (Hankel + Toeplitz) operators. In this paper we solve this problem the space of H.S . operators over the Hilbert space L2(Δ,πα) of square integrable functions over the unit disk Δ equipped with the Dzhrbashyan measure dμ(z) = (α+1)(1- z)αdA(z)(α > -1). This complements the earlier results. In particular we discover many new Ha-plitz type operators. The question of their smoothness properties (Sp- estimates etc.) is however only touched upon

Paracommutators - brief introduction, open problems

Sin resume

The Bellaterra connection

A new object is introduced - the "Fischer bundle" . It is, formally speaking, an Hermitean bundle of infinite rank over a bounded symmetric domain whose fibers are Hilbert spaces whose elements can be realized as entire analytic functions square integrable with respect to a Gaussian measure ("Fischer spaces" ). The definition was inspired by our previous work on the "Fock bundle" . An even more general framework is indicated, which allows one to look upon the two concepts from a unified point of view

Generalizing the arithmatic geometric mean — a hapless computer experiment

The paper discusses the asymptotic behavior of generalizations of the Gauss's arithmetic-geometric mean, associated with the names Meissel (1875) and Borchardt (1876). The ”hapless computer experiment” in the title refers to the fact that the author at an earlier stage thought that one had genuine asymptotic formulae but it is now shown that in general ”fluctuations” are present. However, no very conclusive results are obtained so the paper ends in a conjecture concerning the special rôle of the algorithms of Gauss and Borchardt. The paper discusses the asymptotic behavior of generalizations of the Gauss's arithmetic-geometric mean, associated with the names Meissel (1875) and Borchardt (1876). The ”hapless computer experiment” in the title refers to the fact that the author at an earlier stage thought that one had genuine asymptotic formulae but it is now shown that in general ”fluctuations” are present. However, no very conclusive results are obtained so the paper ends in a conjecture concerning the special rôle of the algorithms of Gauss and Borchardt