Linear Operators: Spectral theory |
From inside the book
Results 1-3 of 90
Page 1187
The inverse of a closed operator is closed . A bounded operator is closed if and only if its domain is closed . PROOF . If A , is the isometric automorphism in H H which maps ( x , y ) into [ y , x ] then r ( T - 1 ) = 4,8 ( T ) which ...
The inverse of a closed operator is closed . A bounded operator is closed if and only if its domain is closed . PROOF . If A , is the isometric automorphism in H H which maps ( x , y ) into [ y , x ] then r ( T - 1 ) = 4,8 ( T ) which ...
Page 1226
Q.E.D. It follows from Lemma 6 ( b ) that any symmetric operator with dense domain has a unique minimal closed symmetric extension . This fact leads us to make the following definition . 7 DEFINITION . The minimal closed symmetric ...
Q.E.D. It follows from Lemma 6 ( b ) that any symmetric operator with dense domain has a unique minimal closed symmetric extension . This fact leads us to make the following definition . 7 DEFINITION . The minimal closed symmetric ...
Page 1393
Let T be a closed operator in Hilbert space . Then the set of complex numbers , such that the range of AI - T is not closed is called the essential spectrum of T and is denoted by 0e ( T ) . It is clear that oe ( T ) Co ( T ) .
Let T be a closed operator in Hilbert space . Then the set of complex numbers , such that the range of AI - T is not closed is called the essential spectrum of T and is denoted by 0e ( T ) . It is clear that oe ( T ) Co ( T ) .
What people are saying - Write a review
We haven't found any reviews in the usual places.
Contents
8 | 876 |
859 | 885 |
extensive presentation of applications of the spectral theorem | 911 |
Copyright | |
44 other sections not shown
Other editions - View all
Common terms and phrases
additive adjoint adjoint operator algebra analytic assume basis belongs Borel set boundary conditions boundary values bounded called clear closed closure commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues elements equal equation equivalent Exercise exists extension fact finite dimensional follows follows from Lemma formal differential operator formula function given Hence Hilbert space Hilbert-Schmidt ideal identity immediately implies independent inequality integral interval invariant isometric isomorphism Lemma limit linear Ly(R mapping matrix measure multiplicity neighborhood norm obtained orthonormal positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown shows solutions spectral spectrum square-integrable statement subset subspace sufficient Suppose symmetric Theorem theory topology transform uniformly unique unit unitary vanishes vector zero