Linear Operators: Spectral theory |
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Page 984
... vanishes in a neighborhood of infinity is dense in L1 ( 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 L1 ( 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 1650
... vanishes in each set I ̧ , in Uala it vanishes PROOF . The proofs of the first four parts of this lemma are left to the reader as an exercise . a To prove ( v ) , we must show from our hypothesis that F ( q ) = 0 if q is in Co ( Uala ) ...
... vanishes in each set I ̧ , in Uala it vanishes PROOF . The proofs of the first four parts of this lemma are left to the reader as an exercise . a To prove ( v ) , we must show from our hypothesis that F ( q ) = 0 if q is in Co ( Uala ) ...
Page 1651
... vanishes outside K , = then it is clear that qyk vanishes outside a compact subset of I - Cp ; thus G ( q ) F ( yk ? ) = 0. This shows that CGC CF , and it is clear conversely that Cp = CGI CCG . This completes the proof of the ...
... vanishes outside K , = then it is clear that qyk vanishes outside a compact subset of I - Cp ; thus G ( q ) F ( yk ? ) = 0. This shows that CGC CF , and it is clear conversely that Cp = CGI CCG . This completes the proof of the ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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A₁ 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 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 open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero