## Linear Operators: Spectral theory |

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Page 876

Then (y-\-Nie)(X) = y(X)+Ni =i(l+N), and

l2/+A^|2 = \(y+Nie)(y+Nie)*\ = \(y+Nie)(y-Nie)\ = \y2+N2e\ ^ ly^+N*. Since this

inequality must hold for all real N, a contradiction is obtained by placing N = \y2\.

Then (y-\-Nie)(X) = y(X)+Ni =i(l+N), and

**hence**|1 + N\ ^ \y + Nie\.**Hence**(1+AO* ^l2/+A^|2 = \(y+Nie)(y+Nie)*\ = \(y+Nie)(y-Nie)\ = \y2+N2e\ ^ ly^+N*. Since this

inequality must hold for all real N, a contradiction is obtained by placing N = \y2\.

Page 1027

5 shows that X is an eigenvalue and

= Xx, and

spectrum of ET. Conversely, suppose that a non-zero scalar X belongs to the ...

5 shows that X is an eigenvalue and

**hence**for some non-zero x in § we have Tx= Xx, and

**hence**, since T = TE, we have (ET)(Ex) = XEx.**Hence**X belongs to thespectrum of ET. Conversely, suppose that a non-zero scalar X belongs to the ...

Page 1227

and 3)_ are clearly linear subspaces of 2)(T*), it remains to show that the spaces

^(T), 25+, and 55_ are mutually orthogonal, and that their sum is ^(T*). Suppose ...

**Hence**T*x = ix, or a;e®+.**Hence**3)+ is closed. Similarly, 5)_ is closed. Since ®+and 3)_ are clearly linear subspaces of 2)(T*), it remains to show that the spaces

^(T), 25+, and 55_ are mutually orthogonal, and that their sum is ^(T*). Suppose ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

Copyright | |

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Acad adjoint extension adjoint operator algebra Amer analytic B-algebra B*-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense differential equations Doklady Akad 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 g Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping Math matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive Proc prove real axis real numbers representation satisfies second order Section sequence singular solution spectral set spectral theory square-integrable subspace Suppose symmetric operator topology transform unique unitary vanishes vector zero