Pro Extensionist Library

By Edwin Bidwell Wilson

ISBN-10: 1172369704

ISBN-13: 9781172369706

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Additional resources for Advanced calculus; a text upon select parts of differential calculus, differential equations, integral calculus, theory of functions; with numerous exercises

Example text

4) T^Tu = u-Pu for all u G V(T). Since T* is a Predholm operator from Hi to H, we can form its generalized inverse (T*)t, which is the linear operator on H to i^i given by (T*) f u = [T* | V{T*) n ^ T * ) 1 ] " 1 ! / - P)u, THEOREM ix G # . 2. If T is a Fredholm operator from H to Hlf then (T+)* = (T*)+. P R O O F . Take arbitrary vectors u e Hi and v e H, and set x = T"^ and = (T*)iv. Clearly x belongs to V{T) nA/'(T) ± and y belongs to £>(T*) nA/'(T*) ± . 3) we have y Tx = u — Qu, T*y = v — Pv, and hence, (Tiu,v) = (x,T*y + Pv) = (Tx,y) = {Tx + Qu, (T*)*v) = {u, (T*)^).

IZ(XI - T) = # . Note that XI — T is also a closed linear operator in H. If 1Z(XI — T) were to equal i7, then by the Closed Graph Theorem (XI — T ) _ 1 would be bounded and A would belong to p(T), a contradiction. 6 (XI — T)~l is unbounded. The set of all such A make up the continuous spectrum of T, which is denoted by ac(T). Case 3. , 7l(XI -T)£H. In this situation the linear opertor (XI — T)~l can be either bounded or unbounded. The set of all such A is called the residual spectrum of T, which we denote by o~r(T).

If T is a Fredholm operator in the Hilbert space H, then the powers Tn, n — 0,1, 2 , . . , and (T*)n, n = 0,1, 2 , . . , are Fredholm operators in H with (T n )* = (T*) n for n = 0 , 1 , 2 , . . In addition, if i(T) = 0, then i{Tn) = z((T*) n ) = 0 for n = 0,1, 2 , . . To conclude this section of basic properties, we turn to the perturbation theory of Fredholm operators. Again let H and Hi be complex Hilbert spaces, and let T be a closed linear operator from H to Hi. We know that the domain V(T) becomes a Hilbert space under the graph norm structure (u,v)T \\u\\T = ( « , i 4 / 2 = [ N | 2 + ||Tw|| 2 ] 1 / 2 .