Linear Operators: Spectral operators |
From inside the book
Results 1-3 of 84
Page 984
... vanishes in a neighborhood of infinity is dense in L ( R ) . PROOF . It follows from Lemma 3.6 that the set of all functions in L2 ( R , B , μ ) which vanish outside of compact sets is dense in this space , and from the Plancherel ...
... vanishes in a neighborhood of infinity is dense in L ( R ) . PROOF . It follows from Lemma 3.6 that the set of all functions in L2 ( R , B , μ ) which vanish outside of compact sets is dense in this space , and from the Plancherel ...
Page 993
... vanishes at the point mo then f is the limit in L1 ( R ) of a sequence { n } each of whose elements has a transform τf vanishing in a neighborhood of mo . PROOF . Let & be the closure of the set of all functions h in L1 ( R ) whose ...
... vanishes at the point mo then f is the limit in L1 ( R ) of a sequence { n } each of whose elements has a transform τf vanishing in a neighborhood of mo . PROOF . Let & be the closure of the set of all functions h in L1 ( R ) whose ...
Page 997
... vanishes on U for every f in L1 ( R ) ~ L¿ ( R ) whose transform vanishes on the complement of V. PROOF . If moo ( q ) then there is a neighborhood V of the identity in R and a neighborhood U of m , such that U ( σ ( p ) + V + V ) is ...
... vanishes on U for every f in L1 ( R ) ~ L¿ ( R ) whose transform vanishes on the complement of V. PROOF . If moo ( q ) then there is a neighborhood V of the identity in R and a neighborhood U of m , such that U ( σ ( p ) + V + V ) is ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
Copyright | |
36 other sections not shown
Other editions - View all
Common terms and phrases
adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers continuous function converges Corollary deficiency indices Definition denote dense eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology unique unitary vanishes vector zero