II 6 Introduction on unbounded linear operators Theorem
II. 6 Introduction on unbounded linear operators
Theorem 3. 3 Riesz Representation Theorem Let X be a Hilbert space and then there is Furthermore the map is conjugate linear and such that
Unbounded linear operator p. 1 E, F: Banach spaces Any linear operator A defined on with range in F is called unbounded linear operator from E to F
Unbounded linear operator p. 2 D(A): domain of A A is called bounded if there is such that
Unbounded linear operator p. 3 G(A)=the graph of A= R(A)=image of A= N(A)=kernel of A=
Unbounded linear operator p. 4 A is called closed if G(A) is closed in i. e. if then
Remark If A is closed, then N(A) is closed in E. In most application, D(A)is dense in E.
Definition of Adjoint A* of A Suppose A is an unbounded linear operator from a Banach space E to a Banach space F with D(A) dense in E Define the adjoint A* of A from F´ to E´ as follows: p. 1
Definition of Adjoint A* of A p. 2 For let By Hahn-Banach Thm, g can be extended to an element of E´. Since D(A) is dense in E, the extension is unique and is denoted by A*v
Definition of Adjoint A* of A p. 3 Obviously, A* is characterized by is linear
Remark If E = F=H is a Hilbert space Identify H´and H by Riesz Representation as a fun. of u is bounded linear on D(A)
Example In If then
Proposition II. 16 Let A: D(A) → F be an unbounded linear operator with dense domain. Then A* is closed i. e. G(A*) is closed in
Ex. F
Remark by then
Remark (26) (27) (28) (29)
Corollary II. 17 be an unbounded Let closed linear operator with then (i) (iii) (iv)
II. 7 Characterization of operators with closed range Surjective operators Bounded operators
Theorem II. 18 be an unbounded Let closed linear operator with then the following properties are equivalent (i) R(A) is closed (ii) R(A*) is closed (iii) (iv)
Theorem II. 19 Let be an unbounded closed linear operator with then the following properties are equivalent (a) A is surjective i. e. R(A)=F (b ) (c)
Theorem II. 19 be an unbounded Let closed linear operator with then (i) (iii) (iv)
Theorem I. 11 Suppose and are convex and suppose that there is such that and is continuous at
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