## Linear Operators: Spectral theory |

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Page 1247

Q.E.D. The next lemma shows that a positive self adjoint transformation has a

transformation, there is a

* = T. PRoof.

Q.E.D. The next lemma shows that a positive self adjoint transformation has a

**unique**positive “square root”. 3 LEMMA. If T is a positive self adjointtransformation, there is a

**unique**positive self adjoint transformation A such that A* = T. PRoof.

Page 1378

matria, measure {6,3, i, j = 1,..., k of Theorem 23 is

p, - 0, if i > k or j > k. PRoof. Suppose that ol, ..., o, is a determining set for T. Then

it is evident from Theorem 23 that if we define {p,}, i, j = 1,..., n, by p,(e) = 6,06), i, ...

matria, measure {6,3, i, j = 1,..., k of Theorem 23 is

**unique**, and 6, = p, i, j = 1,..., k;p, - 0, if i > k or j > k. PRoof. Suppose that ol, ..., o, is a determining set for T. Then

it is evident from Theorem 23 that if we define {p,}, i, j = 1,..., n, by p,(e) = 6,06), i, ...

Page 1383

With boundary conditions A and C, the

boundary condition raq = Ao is sin Vit. With boundary conditions A, the

eigenvalues are consequently to be determined from the equation sin V2 = 0.

With boundary conditions A and C, the

**unique**solution of tao = Ao satisfying theboundary condition raq = Ao is sin Vit. With boundary conditions A, the

eigenvalues are consequently to be determined from the equation sin V2 = 0.

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

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adjoint extension adjoint operator algebra Amer analytic B-algebra Banach Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients complete complex numbers 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 f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping Math matrix measure Nauk SSSR N.S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Plancherel's theorem positive Proc PRoof prove real numbers satisfies sequence singular ſº solution spectral spectral set spectral theory square-integrable subspace Suppose theory To(r topology transform unique unitary vanishes vector zero