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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.102 Introduction to Functional Analysis Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 104 LECTURE NOTES FOR 18.102, SPRING 2009 Lecture 18. Tuesday April 14: Compact operators Last time we considered invertible elemenets of B ( H ) , the algebra of bounded operators on a separable Hilbert space, and also the finite rank operators. The latter form an ideal which is closed under taking adjoints. We also showed the the closure of this ideal, the elements in B ( H ) which are the limits of (normconvergent) sequences of finite rank operators, also form an ideal which is closed under taking adjoints and also norm, i.e. metrically, closed. Definition 8 . An element K ∈ B ( H ) , the bounded operators on a separable Hilbert space, is said to be compact (the old terminology was ‘totally bounded’ and you might still see this) if the image of the unit ball is precompact, i.e. has compact closure – that is if the closure of K { u ∈ H ; u H ≤ 1 } is compact in H . Lemma 12. An operator K ∈ B ( H ) is compact if and only if the image { Ku n } of any weakly convergent sequence { u n } in H is strongly, ie. norm, convergent. Proof. First suppose that u n u is a weakly convergent sequence in H and that K is compact. We know that u n < C is bounded so the sequence Ku n is contained in CK ( B (0 , 1)) and hence in a compact set (clearly if D is compact then so is cD for any constant c. ) Thus, any subsequence of Ku n has a convergent subseqeunce and the limit is necessarily Ku since Ku n Ku (true for any bounded operator by computing (18.1) ( Ku n ,v ) = ( u n ,K ∗ v ) ( u,K ∗ v ) = ( Ku,v ) . ) → But the condition on a sequence in a metric space that every subsequence of it has a subsequence which converges to a fixed limit implies convergence. (If you don’t remember this, reconstruct the proof: To say a sequence v n does not converge to...
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This note was uploaded on 11/08/2011 for the course PHY 18.102 taught by Professor Staff during the Spring '09 term at MIT.
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