Linear Operators: Spectral theory |
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Page 1174
... f ( s ) ds = [ h ( s ) \ f ( s ) \ ds ; hence ( 42 ) follows from the similar well - known equation for scalar- valued functions . This concludes the proof of Lemma 21 , and with it the proof of Theorem 20. Q.E.D. 22 COROLLARY . Let ( S ...
... f ( s ) ds = [ h ( s ) \ f ( s ) \ ds ; hence ( 42 ) follows from the similar well - known equation for scalar- valued functions . This concludes the proof of Lemma 21 , and with it the proof of Theorem 20. Q.E.D. 22 COROLLARY . Let ( S ...
Page 1649
... let F be a distribution in I. Then the distribution F in I defined by the equation F ( q ) = F ( 9 ) , q Є Co ( I ) , is called the complex conjugate of F. 8 LEMMA . Let I and F be as in the preceding definition , and let t be a formal ...
... let F be a distribution in I. Then the distribution F in I defined by the equation F ( q ) = F ( 9 ) , q Є Co ( I ) , is called the complex conjugate of F. 8 LEMMA . Let I and F be as in the preceding definition , and let t be a formal ...
Page 1662
... F DEFINITION . Let I be an open subset of C , and let F be in D ( I ) . Then the closed set C in I , which is the complement of the largest open set in I in which F vanishes , i.e. , which is the complement in I of the union of all the ...
... F DEFINITION . Let I be an open subset of C , and let F be in D ( I ) . Then the closed set C in I , which is the complement of the largest open set in I in which F vanishes , i.e. , which is the complement in I of the union of all the ...
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