## Linear Operators: Spectral operators |

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Page 1105

But |tr(D, U, D, U2 Da Ua) is |D|1|Dal-Da'.

and Corollary 8. Q.E.D. We may now proceed rapidly to the main goal of the

present section: the derivation of inequalities for the resolvent of an operator in C,

...

But |tr(D, U, D, U2 Da Ua) is |D|1|Dal-Da'.

**follows immediately**from Lemma 9(d)and Corollary 8. Q.E.D. We may now proceed rapidly to the main goal of the

present section: the derivation of inequalities for the resolvent of an operator in C,

...

Page 1226

Part (a)

from part (a) and Lemma 5(c). Q.E.D. It follows from Lemma 6(b) that any

symmetric operator with dense domain has a unique minimal closed symmetric

extension.

Part (a)

**follows immediately**from Lemma 5(b), and part (b)**follows immediately**from part (a) and Lemma 5(c). Q.E.D. It follows from Lemma 6(b) that any

symmetric operator with dense domain has a unique minimal closed symmetric

extension.

Page 1231

... where & Co, G \ , then &^ = (31)*. PRoof. Statement (a)

from Definition 14 and Definition 4. Statement (b)

Definition 14 and Lemma 15. By Lemma 8(c) {a, y} = 0 for a in Q(T*) and ...

... where & Co, G \ , then &^ = (31)*. PRoof. Statement (a)

**follows immediately**from Definition 14 and Definition 4. Statement (b)

**follows immediately**fromDefinition 14 and Lemma 15. By Lemma 8(c) {a, y} = 0 for a in Q(T*) and ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

52 other sections not shown

### 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