Open Access
2011 The Polar Decomposition in Banach Spaces
T. L. Gill, V. Steadman, W. W. Zachary
Afr. Diaspora J. Math. (N.S.) 11(2): 98-131 (2011).


In this paper we survey research progress related to the existence of an adjoint for linear operators on Banach spaces. We introduce a new pair separable Banach spaces which are required for the general theory. We then discuss a number ways one can explicitly construct an adjoint and then prove that one always exists for bounded linear operators. However, this is not true for the class of closed densely defined linear operators. In this case, we can only show that one exists for operators of Baire class one. The existence of an adjoint allows us to construct the polar decomposition. As applications, we extend the Poincaré inequality and the Stone-von Neumann version of the spectral theorem to all operators of Baire class one on a separable Banach space. Our results even show that the spectral theorem is natural for Hilbert spaces (in a certain well-defined sense). As a final application, we provide the natural Banach space version of the Schatten class of compact operators.


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T. L. Gill. V. Steadman. W. W. Zachary. "The Polar Decomposition in Banach Spaces." Afr. Diaspora J. Math. (N.S.) 11 (2) 98 - 131, 2011.


Published: 2011
First available in Project Euclid: 6 December 2011

zbMATH: 1267.47001
MathSciNet: MR2862568

Primary: 46B03
Secondary: 47D03 , 47F05 , 47H06

Keywords: Poincaré inequality spectral theorem , Schatten-class , semigroups , vector measures , vector-valued functions

Rights: Copyright © 2011 Mathematical Research Publishers

Vol.11 • No. 2 • 2011
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