Linear Operators, Part 2 |
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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 997
... vanishes on U for every ƒ 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 ƒ 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 ...
Page 1651
... vanishes outside K , then it is clear that qyk vanishes outside a compact subset of I - CF ; thus G ( q ) F ( x ) = 0. This shows that CGCC , and it is clear conversely that Cp = CGI CCG . This completes the proof of the existence of G ...
... vanishes outside K , then it is clear that qyk vanishes outside a compact subset of I - CF ; thus G ( q ) F ( x ) = 0. This shows that CGCC , and it is clear conversely that Cp = CGI CCG . This completes the proof of the existence of G ...
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BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 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 real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero