Unbounded linear operators in seminormed spaces

Master Thesis


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University of Cape Town

Linear operator theory is usually studied in the setting of normed or Banach spaces. However, careful examination of proofs shows that in many cases the Hausdorff property of normed spaces is not used. Even in those cases where explicit use of the Hausdorff property is made, one can often get around this (should one wish to work in seminormed spaces) by suitable identification of elements and then working in the resulting normed space. Working in seminormed spaces rather than normed spaces is especially advantageous when dealing with quotients (which occur in linear operator theory when one considers the factorisation of an operator through its domain space quotiented by its null space): when taking the quotient of a normed space by a subspace, one requires the subspace to be closed in order for the quotient to be a normed space; however, in the seminormed space case the requirement that the subspace be closed is no longer necessary. Seminorms are also important in the study of certain properties of the second adjoint of an operator (for example, seminorms occur in the study of operators of the Tauberian type (see [C2]) and operators analagous to weakly compact operators (see Chapter VI). It is the aim of this work to generalise as much of the basic theory of unbounded linear operators as possible to seminormed spaces. In Chapter I, some aspects of topological vector spaces (which will be used throughout this work) are presented, the most important parts being the Hahn-Banach theorem and the section on weak topologies. In Chapter II, we restrict our attention to seminormed spaces, the setting in which the remainder of this work takes place. The basic theory of unbounded linear operators, their adjoints and the relationship between operators and their adjoints is covered in Chapter III. Chapter IV concentrates on characterising unbounded strictly singular operators while in Chapter V operators with closed range are studied. Finally, in Chapter VI, a property corresponding · to one of the equivalent conditions for a bounded operator to be weakly compact is studied for unbounded operators.

Bibliography: pages 101-104.