A Class of Toeplitz Operators in Several Complex Variables

Abstract

In order to study the Toeplitz algebras related to a Dirac operators in a neighborhood of a closed bounded domain D with smooth boundary in C n we introduce a singular Cauchy type integral. We compute its principal symbol, thus initiating the index theory.Keywords: Dirac operators, Cauchy type integral, symbol, Toeplitz operators, index. DOI: 10.17516/1997-1397-2017 There are a number of ways in which the theory of Toeplitz operators can be generalised to n dimensions, see e.g. This work focuses on a new class of Toeplitz operators which is more closely related to elliptic theory. The new Toeplitz operators admit very transparent description which motivates strikingly their study. To this end, let A be a (k×k) -matrix of first order scalar partial differential operators in a neighborhood of the closed bounded domain D with smooth boundary S in C n . Assume that the leading symbol of A has rank k away from the zero section of the cotangent bundle of D. Then, given any solution u of the homogeneous equation Au = 0 in the interior of D which has finite order of growth at the boundary, the Cauchy data t(u) of u with respect to A possess weak limit values at the boundary. If A satisfies the so-called uniqueness condition of the local Cauchy problem in a neighborhood of D, then the solution u is uniquely defined by its Cauchy data at S. Let t(u) = Bu be a representation of the Cauchy data of u. The space of all Cauchy data Bu of u at the boundary is effectively described by the condition of orthogonality to solutions of the formal adjoint equation A * g = 0 near D by means of a Green formula, see [6, § 10.3.4]. In this way we distinguish Hilbert space of vector-valued functions on S which represent solutions to Au = 0 in the interior of D. In particular, one introduces Hardy spaces H as subspaces of L 2 (S, C k ) consisting of the Cauchy data of solutions to Au = 0 in the interior of D with appropriate behaviour at the boundary. Pick such a Hilbert space H. By the above, H is a closed subspace of L 2 (S, C k ) and we write Π for the orthogonal projection of L 2 (S, C k ) onto H. If A is the Cauchy-Riemann operator then Π just amounts to the Szegö projection.Given a (k × k) -matrix M (z) of smooth function on S, the operator T M on H given by u → Π(M u) is said to be a Toeplitz operator with multiplier M . If M is a scalar multiple of the * fdp@bk.ru c ⃝ Siberian Federal University. All rights reserved -206 -