## Linear Operators: Spectral theory |

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Page 1190

Q.E.D. Most of the considerations in this chapter and the next will be directed

towards an

following definition. 7 Definition. The

(x, Ty) ...

Q.E.D. Most of the considerations in this chapter and the next will be directed

towards an

**operator**which is either**symmetric**or self adjoint according to thefollowing definition. 7 Definition. The

**operator**T is said to be**symmetric**if (Tx, y) =(x, Ty) ...

Page 1223

In the theory of bounded

T is everywhere defined and

situation is quite different. Consider, as an example, an

In the theory of bounded

**operators**, we have only to verify**symmetry**(T* D T), for ifT is everywhere defined and

**symmetric**, then T* = T. But if T is unbounded thesituation is quite different. Consider, as an example, an

**operator**which will be ...Page 1272

Maximal

then it has proper symmetric extensions provided both of its deficiency indices

are different from zero. A maximal

...

Maximal

**symmetric operators**. If T is a**symmetric operator**with dense domain,then it has proper symmetric extensions provided both of its deficiency indices

are different from zero. A maximal

**symmetric operator**is one which has no proper...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

Copyright | |

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### Common terms and phrases

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