Mathematical Sciences: "Applications of Extension Theorems on Krein Spaces"

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This research explores various applications of the extension theorems and commutant lifting theorems for contraction operators on Krein spaces. Some of these problems, such as the Krein space formulation of Foias' generalization of an interpolation problem of Dym and Gohberg, are natural restatements of classical Hilbert space problems. Other of these problems, such as the study of maximal negative subspaces of bicontractive operators, make sense only within the context of indefinite inner product spaces. This research is in the general area of modern analysis. Its focus is on the commutant lifting theorems for certain classes of operators on Krein spaces. For Hilbert space contractions, the lifting theorem has been well known since the late 1960's and has been a powerful tool in areas such as classical interpolation theory and in corona theory.