Integral Operators

Leif Mejlbro Integral Operators Download free ebooks at bookboon.com 2 Integral Operators © 2009 Leif Mejlbro & Ventus Publishing ApS ISBN 978-87-7681-529-5 Disclaimer: The texts of the advertisements are the sole responsibility of Ventus Publishing, no endorsement of them by the author is either stated or implied. Download free ebooks at bookboon.com 3 Integral Operators Contents Contents 1. Hilbert-Smith operators 5 2. Other types of integral operators 47 Index 66 The next step for top-performing graduates Masters in Management Designed for high-achieving graduates across all disciplines, London Business School’s Masters in Management provides specific and tangible foundations for a successful career in business. This 12-month, full-time programme is a business qualification with impact. In 2010, our MiM employment rate was 95% within 3 months of graduation*; the majority of graduates choosing to work in consulting or financial services. As well as a renowned qualification from a world-class business school, you also gain access to the School’s network of more than 34,000 global alumni – a community that offers support and opportunities throughout your career. For more information visit www.london.edu/mm, email mim@london.edu or give us a call on +44 (0)20 7000 7573. * Figures taken from London Business School’s Masters in Management 2010 employment report Download free ebooks at bookboon.com 4 Integral Operators 1. Hilbert-Smith operators 1 Hilbert-Schmidt operators Example 1.1 Let (ek) denote an orthonormal basis in a Hilbert space H, and assume that the operator T has the matrix representation (tjk) with respect to the basis (ek). Show that XX |tjk| < ∞ j=1 k=1 implies that T is compact. Let (fk) denote another orthonormal basis in H, and let sjk = (Tfj,fk) so that (sjk) is the matrix representation of T with respect to the basis (fk). Show that XX XX |tjk| = |sjk| . j=1 k=1 j=1 k=1 An operator satisfying XX |tjk| < ∞ j=1 k=1 is called a general Hilbert-Schmidt operator. Write tjk = (Tej,ej). It follows from Ventus, Hilbert spaces, etc., Example 2.7 that ⎛ ⎞ +∞ +∞ X Tx = T xjej = xjtjkek. j=1 j=1 k=1 Define the sequence (Tn) of operators by ⎛ ⎞ +∞ +∞ X Tnx = Tn xjej = xjtjkek. j=1 j=1 k=1 The range of Tn is finite dimensional, so Tn is compact. Then we conclude from +∞ X 2 +∞ +∞ 2 k(T −Tn)xk = xjtjkek = xjtjk , j=1 n=1 k=n+1 j=1 where ⎧ ⎫ ⎧ ⎫ +∞ +∞ +∞ xjtjk ≤ |xj|2 · |tjk|2 , j=1 j=1 j=1 Download free ebooks at bookboon.com 5 ... - tailieumienphi.vn