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 , u ) 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 , u ) 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 To prove ( v ) , we must show from our hypothesis that F ( q ) = 0 if q is in Co ( Uala ) ...
... 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 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 pyk vanishes outside a compact subset of I - Cp ; thus G ( p ) F ( x ) = 0 . This shows that CCCp , and it is clear conversely that C = CGICCG . This completes the proof of the existence of G ...
... vanishes outside K , then it is clear that pyk vanishes outside a compact subset of I - Cp ; thus G ( p ) F ( x ) = 0 . This shows that CCCp , and it is clear conversely that C = CGICCG . This completes the proof of the existence of G ...
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BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero