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 , it vanishes in Uala PROOF . The proofs of the first four parts of this lemma are left to the reader as an exercise . α α To prove ( v ) , we must show from our hypothesis that F ( q ) = 0 if q is in Co ( UaIa ) ...
... vanishes in each set I , it vanishes in Uala PROOF . The proofs of the first four parts of this lemma are left to the reader as an exercise . α α To prove ( v ) , we must show from our hypothesis that F ( q ) = 0 if q is in Co ( UaIa ) ...
Page 1651
Nelson Dunford, Jacob T. Schwartz. and vanishes outside K , then - vanishes outside a compact Ф subset of I - Cp , so that G ( ) = F ( ¥ Ê ¥ ) = F ( q ) . Thus GI = F. If KC = 0 and the function in Co ( II ) vanishes outside K , then it ...
Nelson Dunford, Jacob T. Schwartz. and vanishes outside K , then - vanishes outside a compact Ф subset of I - Cp , so that G ( ) = F ( ¥ Ê ¥ ) = F ( q ) . Thus GI = F. If KC = 0 and the function in Co ( II ) vanishes outside K , then it ...
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BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero