Integral Operators Functional Analysis Examples c-5 Leif Mejlbro Download free books at Leif Mej lbro I nt egral Operat ors Download free eBooks at bookboon.com I nt egral Operat ors © 2009 Leif Mej lbro & Vent us Publishing ApS I SBN 978- 87- 7681- 529- Disclaim er: The t ext s of t he advert isem ent s are t he sole responsibilit y of Vent us Publishing, no endorsem ent of t hem by t he aut hor is eit her st at ed or im plied Download free eBooks at bookboon.com Contents Integral Operators Cont ent s Hilbert-Smith operators Other types of integral operators Index 47 66 www.sylvania.com We not reinvent the wheel we reinvent light Fascinating lighting offers an ininite spectrum of possibilities: Innovative technologies and new markets provide both opportunities and challenges An environment in which your expertise is in high demand Enjoy the supportive working atmosphere within our global group and beneit from international career paths Implement sustainable ideas in close cooperation with other specialists and contribute to inluencing our future Come and join us in reinventing light every day Light is OSRAM Click on the ad to read more Download free eBooks at bookboon.com Hilbert-Smith operators Integral Operators 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 ∞ ∞ |tjk | < ∞ j=1 k=1 implies that T is compact Let (fk ) denote another orthonormal basis in H, and let sjk = (T fj , fk ) so that (sjk ) is the matrix representation of T with respect to the basis (fk ) Show that ∞ ∞ ∞ ∞ 2 |sjk | |tjk | = j=1 k=1 j=1 k=1 An operator satisfying ∞ ∞ |tjk | < ∞ j=1 k=1 is called a general Hilbert-Schmidt operator Write tjk = (T ej , ej ) It follows from Ventus, Hilbert spaces, etc., Example 2.7 that ⎞ ⎛ +∞ +∞ +∞ Tx = T ⎝ j=1 xj ej ⎠ = xj tjk ek j=1 k=1 Define the sequence (Tn ) of operators by ⎞ ⎛ +∞ n +∞ Tn x = T n ⎝ j=1 xj ej ⎠ = xj tjk ek j=1 k=1 The range of Tn is finite dimensional, so Tn is compact Then we conclude from 2 +∞ +∞ (T − Tn ) x = +∞ xj tjk ek j=1 n=1 +∞ = xj tjk , k=n+1 j=1 where +∞ xj tjk j=1 ≤ ⎧ ⎨+∞ ⎩ j=1 ⎫ ⎧ ⎫ ⎬ ⎨+∞ ⎬ 2 |xj | |tjk | · , ⎭ ⎩ ⎭ j=1 Download free eBooks at bookboon.com Hilbert-Smith operators Integral Operators that (T − Tn ) x ≤ ⎧ ⎨ ⎩ +∞ +∞ |tjk | k=n+1 j=1 ⎫ ⎬ · x ⎭ It follows that +∞ T − Tn +∞ ≤ |tjk | k=n+1 j=1 Putting +∞ |tjk | ≥ 0, ak = j=1 it follows from the assumption that +∞ +∞ +∞ |tjk | < +∞ ak = k=1 j=1 k=1 Hence, to every ε > there is an n ∈ N, such that +∞ ak < ε2 , k=n+1 360° thinking Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities Click on the ad to read more Download free eBooks at bookboon.com Hilbert-Smith operators Integral Operators from which +∞ T − Tn +∞ +∞ ≤ ak < ε2 , |tjk | = k=n+1 j=1 k=n+1 thus T − Tn < ε, and we have proved that Tn → T Because all the Tn are compact, we conclude that T is also compact Given another orthonormal basis (fk ) of H, and let sjk = (T fj , fk ) Then an application of Parseval’s equation gives that |(T ek , fj )| = j=1 k=1 +∞ +∞ +∞ +∞ +∞ +∞ +∞ T ej |tkj | j=1 k=1 k=1 j=1 k=1 2 |(T ek , ej )| = = and |(T ek , fj )| 2 T ⋆ fj |(ek , T ⋆ fj )| = = |(T ⋆ fj , fk )| = |sjk | , |(T fj , fk )| = j=1 k=1 j=1 k=1 2 |(fj , T fk )| = +∞ +∞ +∞ +∞ +∞ +∞ = j=1 k=1 j=1 j=1 k=1 j=1 k=1 +∞ +∞ +∞ +∞ +∞ +∞ +∞ j=1 k=1 hence, +∞ +∞ +∞ +∞ +∞ +∞ |sjk | |tkj | = j=1 k=1 j=1 k=1 j=1 k=1 2 |tjk | = Example 1.2 For a general Hilbert-Schmidt operator we define the Hilbert-Schmidt norm ⎧ ⎨+∞ +∞ T HS = ⎩ |tjk | ⎫ 21 ⎬ ⎭ j=1 k=1 Show that this is a norm, and show that T ≤ T HS for a general Hilbert-Schmidt operator T Write tjk = (T ej , ek ), and let ⎧ ⎨+∞ +∞ T HS = ⎩ |tjk | j=1 k=1 ⎫ 12 ⎬ ⎭ Then T HS ≥ 0, and if T HS = 0, then tjk = (T ej , ek ) = for all j, k ∈ N, thus +∞ +∞ tjk ek = (T ej , ek ) ek = T ej = k=1 for every j ∈ N k=1 Download free eBooks at bookboon.com · HS by ... aut hor is eit her st at ed or im plied Download free eBooks at bookboon.com Contents Integral Operators Cont ent s Hilbert-Smith operators Other types of integral operators Index 47 66 www.sylvania.com... eBooks at bookboon.com Hilbert-Smith operators Integral Operators Hilbert-Schmidt operators Example 1.1 Let (ek ) denote an orthonormal basis in a Hilbert space H, and assume that the operator... sjk = (T fj , fk ) so that (sjk ) is the matrix representation of T with respect to the basis (fk ) Show that ∞ ∞ ∞ ∞ 2 |sjk | |tjk | = j=1 k=1 j=1 k=1 An operator satisfying ∞ ∞ |tjk | < ∞