Complex Analysis and Operator Theory

, Volume 6, Issue 2, pp 491–513 | Cite as

The Szegö Metric Associated to Hardy Spaces of Clifford Algebra Valued Functions and Some Geometric Properties

  • Dennis Grob
  • Rolf Sören KraußharEmail author


In analogy to complex function theory we introduce a Szegö metric in the context of hypercomplex function theory dealing with functions that take values in a Clifford algebra. In particular, we are dealing with Clifford algebra valued functions that are annihilated by the Euclidean Dirac operator in \({\mathbb{R}^{m+1}}\) . These are often called monogenic functions. As a consequence of the isometry between two Hardy spaces of monogenic functions on domains that are related to each other by a conformal map, the generalized Szegö metric turns out to have a pseudo-invariance under Möbius transformations. This property is crucially applied to show that the curvature of this metric is always negative on bounded domains. Furthermore, it allows us to establish that this metric is complete on bounded domains.


Bounded Domain Dirac Operator Hardy Space Bergman Space Clifford Algebra 
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© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Lehrstuhl A für MathematikRheinisch-Westfälische Technische Hochschule AachenAachenGermany
  2. 2.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany

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