## Linear Operators: Spectral operators |

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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

= (a, ...

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 (Tr, y)= (a, ...

Page 1223

not tht ord s In the theory of bounded

T*D T), for if T is everywhere defined and

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

...

not tht ord s In the theory of bounded

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

**symmetric**, then To = T. But if T isunbounded the situation is quite different. Consider, as an example, an

**operator**...

Page 1272

Marimal

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

are different from zero. A marimal

...

Marimal

**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 marimal

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

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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### Common terms and phrases

adjoint extension adjoint operator algebra analytic B-algebra B-space Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients compact operator complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality infinity integral interval kernel Lemma Let f Let Q linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real numbers satisfies sequence singular ſº solution spectral spectral theorem square-integrable subspace Suppose symmetric operator theory To(r To(t topology transform uniformly unique unitary vanishes vector zero