## Linear Operators: Spectral operators |

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Page 1272

If T is a symmetric operator with dense domain, then it has proper symmetric

extensions provided both of its

marimal symmetric operator is one which has no proper symmetric extensions;

hence, ...

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. Amarimal symmetric operator is one which has no proper symmetric extensions;

hence, ...

Page 1295

Definition XII.4.9) that if t is formally self adjoint, the positive and negative

deficiency spaces of To(t) are the manifolds Q ... If t is a formally self adjoint

formal differential operator of order n, both

than or equal to ...

Definition XII.4.9) that if t is formally self adjoint, the positive and negative

deficiency spaces of To(t) are the manifolds Q ... If t is a formally self adjoint

formal differential operator of order n, both

**deficiency indices**of To(t) are lessthan or equal to ...

Page 1454

If T is a closed symmetric operator in Hilbert space, and T is bounded below, then

(a) the essential spectrum of T is a subset of the real aris which is bounded below

; (b) the

If T is a closed symmetric operator in Hilbert space, and T is bounded below, then

(a) the essential spectrum of T is a subset of the real aris which is bounded below

; (b) the

**deficiency indices**of T are equal. PRoof. To prove (a), note that if T is ...### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### 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