D. , T ) is called symmetric if T ? T *

D. , T ) is called self-adjoint if T = T *

, A symmetric operator is essentially self-adjoint if its closure is self-adjoint

, In the above definition, we used the relation A ? B between two operators A and B. It means that Dom(A) ? Dom(B)

, T ) is essentially self-adjoint operator, it has unique self-adjoint extension

, We often use the following theorem to produce a self-adjoint operator from a continuous and coercive sesquilinear

V. and ·. , be a Hilbert space such that V is continuously embedded and dense in H. Let Q be a sesquilinear form define on V

. V--elliptic-(or-coercive, There exists a constant ? >

. Then,

. Furthermore, L is bijective from Dom(S) onto H and Dom(S) is dense in V and in H

, Definition A.7 (Resolvent and spectrum set). Let (Dom(T ), T ) be a self-adjoint operator on H. The resolvent set of T is define by ?(T ) = {z ? C : (T ? zI) is a bijective from Dom(T ) onto H}

, and the complement of the resolvent set in C is called the spectrum of the operator, Sp(T ) = C \ ?(T )

, And if ? ? ?(T ), the operator (T ? ?) ?1 is called a resolvent of T

, T ) is a self-adjoint operator, its spectrum is classified into discrete spectrum and essential spectrum: Definition A.8. Let (Dom(T ), T ) be a self

, The discrete spectrum of T , denoted by Sp dis , containing elements which are isolated finite multiplicity values in Sp(T )

, The essential spectrum of T , denoted by Sp ess (T ), is the complement of discrete spectrum of T Sp ess (T ) = Sp(T ) \ Sp dis (T )

, The operator T will have a purely discrete spectrum if it has a compact resolvent

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