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Phone: (407) 823-6284;   Fax: (407) 823-6253;   MAP  207 2000-2001 MATHEMATICS COLLOQUIUM SERIES 02/20/01 Colloquium PROFESSOR LANCE L. LITTLEJOHN Department of Mathematics and Statistics Utah State University Left-Definite Spectral Theory for a Class of Positive Self-Adjoint Operators Abstract: In this lecture, we show that any self-adjoint operator A in a Hilbert space H that is bounded below by a positive constant generates a continuum of Hilbert spaces and a continuum of self-adjoint operators . For reasons originating in differential equations, we call left-definite space and left-definite operator associated with the pair (H,A). The Hilbert space spectral theorem plays a fundamental role in these constructions. One of the surprising consequences of this general theory is the fact that new information can be obtained about the original operator A. Besides describing this general left-definite theory, we illustrate it by considering several examples.