Linear Operators: Spectral theory |
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Page 1653
... Statement ( i ) follows from statement ( ii ) by Definitions 15 ( iii ) and 17 ( ii ) . Statement ( iii ) follows from statement ( ii ) and the fact that F ( +1 ) F ( x ) for all k≥0 and Fin H ( +1 ) ( I ) , ( cf. Definition 15 ( i ) ...
... Statement ( i ) follows from statement ( ii ) by Definitions 15 ( iii ) and 17 ( ii ) . Statement ( iii ) follows from statement ( ii ) and the fact that F ( +1 ) F ( x ) for all k≥0 and Fin H ( +1 ) ( I ) , ( cf. Definition 15 ( i ) ...
Page 1756
... statement ( i ) . Moreover , statement ( i ) enables us to reduce the proof of the existence of the function V to the proof of the following statement . ( ii ) For each r > 0 and p ≥ 1 , there exists a function Vin ĈP ( E ) , such that ...
... statement ( i ) . Moreover , statement ( i ) enables us to reduce the proof of the existence of the function V to the proof of the following statement . ( ii ) For each r > 0 and p ≥ 1 , there exists a function Vin ĈP ( E ) , such that ...
Page 1760
... Statement ( vii ) can be deduced from statement ( viii ) as follows . Let k ' in ( viii ) be set equal to k + v + 1 , where v = [ ( n + 1 ) / 2 ] and kis as in ( vii ) . Let λ and a be related by a = e- " , so that " a sufficiently ...
... Statement ( vii ) can be deduced from statement ( viii ) as follows . Let k ' in ( viii ) be set equal to k + v + 1 , where v = [ ( n + 1 ) / 2 ] and kis as in ( vii ) . Let λ and a be related by a = e- " , so that " a sufficiently ...
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