## Linear Operators: Spectral operators |

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Page 1019

Nelson Dunford, Jacob T. Schwartz. volume of any n-sided parallelepiped is

never larger than the volume of a rectangular parallelepiped with sides of the

same lengths. PRoof. The

checked for ...

Nelson Dunford, Jacob T. Schwartz. volume of any n-sided parallelepiped is

never larger than the volume of a rectangular parallelepiped with sides of the

same lengths. PRoof. The

**inequality**is obvious for n = 1 and may be easilychecked for ...

Page 1094

Statement (a) is evident from Definition 1. To prove (b), we put T = TI--T, and note

that, by Corollary 3, Aug. 11(T1-HT2) is un, 1(T1)+u,11(Ts) (12,12(Ti-HT2) is un, 1

(T)+/1,12(T2). First let p > 1. Then by Minkowski's

Statement (a) is evident from Definition 1. To prove (b), we put T = TI--T, and note

that, by Corollary 3, Aug. 11(T1-HT2) is un, 1(T1)+u,11(Ts) (12,12(Ti-HT2) is un, 1

(T)+/1,12(T2). First let p > 1. Then by Minkowski's

**inequality**, 1/p (So....(t))".<|x||.Page 1105

By the continuity of tr(T) for Te C1, the continuity of the product TS which was

noted in the paragraph following Lemma 9, the continuity of the norm function

which follows from the triangle

follows ...

By the continuity of tr(T) for Te C1, the continuity of the product TS which was

noted in the paragraph following Lemma 9, the continuity of the norm function

which follows from the triangle

**inequality**of Lemma 14(d), and by Lemma 11, itfollows ...

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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