Linear Operators: Spectral theory |
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Page 984
... dense in this space , and from the Plancherel theorem that the set of all ƒ in L2 ( R ) for which f vanishes except on a compact set in R is dense in L2 ( R ) . Since the map [ f , g ] → fg takes L2 ( R ) × L2 ( R ) onto all of L1 ( R ) ...
... dense in this space , and from the Plancherel theorem that the set of all ƒ in L2 ( R ) for which f vanishes except on a compact set in R is dense in L2 ( R ) . Since the map [ f , g ] → fg takes L2 ( R ) × L2 ( R ) onto all of L1 ( R ) ...
Page 1245
... dense domain in Hilbert space has a unique factorization T PA , where A is a positive ( i.e. , ( Ax , x ) ≥ 0 , x ... dense domain . Then tion ; ( a ) D ( T ) is dense and T ** = T ; ( b ) ( I + T * T ) -1 exists and is a bounded self ...
... dense domain in Hilbert space has a unique factorization T PA , where A is a positive ( i.e. , ( Ax , x ) ≥ 0 , x ... dense domain . Then tion ; ( a ) D ( T ) is dense and T ** = T ; ( b ) ( I + T * T ) -1 exists and is a bounded self ...
Page 1905
... Dense convex sets , V.7.27 ( 437 ) Dense linear manifolds , V.7.40-41 ( 438-439 ) Dense set , definition , I.6.11 ( 21 ) density of continuous functions in TM and L ,, III.9.17 ( 170 ) , IV.8.19 ( 298 ) density of simple functions in L ...
... Dense convex sets , V.7.27 ( 437 ) Dense linear manifolds , V.7.40-41 ( 438-439 ) Dense set , definition , I.6.11 ( 21 ) density of continuous functions in TM and L ,, III.9.17 ( 170 ) , IV.8.19 ( 298 ) density of simple functions in L ...
Contents
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