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Continuous Linear Monotone Operators on Banach Spaces

Bauschke, Heinz H. and Borwein, Jonathan M. (1995) Continuous Linear Monotone Operators on Banach Spaces. [Preprint]

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      Abstract

      The concept of a monotone operator --- which covers both linear positive semi-definite operators and subdifferentials of convex functions --- has turned out to be very powerful in various branches of mathematics. Over the last few decades, several new notions of monotonicity have been introduced: Gossez' maximal monotone of type (D), Simons' monotone of type (WD) and of type (NI), Fitzpatrick and Phelps' locally maximal monotone. While these monotonicities are automatic for maximal monotone operators in reflexive Banach spaces and for subdifferentials of convex functions, their precise relationship is largely unknown. In view of the origin of the theory of monotone operators, it is very natural to investigate linear monotone (i.e. positive semi-definite) operators. Here, it is shown --- within the beautiful framework of Convex Analysis --- that for continuous linear monotone operators, {\em all these notions coincide and are equivalent to the monotonicity of the conjugate operator.} The latter condition is analyzed and illustrated by several examples. Some nonlinear results on regularizations conclude the paper.

      Item Type: Preprint
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: positive semi-definite operator, antisymmetric operator, skew-symmetric operator, symmetric operator, monotone operator, monotone operator of type (D), locally maximal monotone operator, weakly compact operator, property (w), duality mapping, regularization
      Subjects: 46-xx Functional analysis > 46Bxx Normed linear spaces and Banach spaces; Banach lattices
      26-xx Real functions > 26Axx Functions of one variable
      47-xx Operator theory > 47Bxx Special classes of linear operators
      47-xx Operator theory > 47Hxx Nonlinear operators and their properties
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 17 Nov 2003
      Last Modified: 21 Apr 2010 11:13
      URI: https://docserver.carma.newcastle.edu.au/id/eprint/114

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