## Linear Operators: Spectral operators |

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Page 940

Let f be a

defined on G by the equation f'(t) = f(ts). The function f° is the translate of f by s.

The function j is said to be finite dimensional if its set {f's e G} of translates is a

finite ...

Let f be a

**complex valued**function defined on a group G. For s in G let fo bedefined on G by the equation f'(t) = f(ts). The function f° is the translate of f by s.

The function j is said to be finite dimensional if its set {f's e G} of translates is a

finite ...

Page 980

It should be recalled (IX.2.2) that there is a one-to-one correspondence between

the non-zero

and the maximal ideals in Qs. This correspondence is given by the equation ...

It should be recalled (IX.2.2) that there is a one-to-one correspondence between

the non-zero

**complex valued**homomorphisms on any commutative B-algebra QIand the maximal ideals in Qs. This correspondence is given by the equation ...

Page 1281

Let t be a formal differential operator of order n on the interval I. Suppose that g is

a measurable

subinterval of I. Let to e I, and let co, c1, ..., c, 1 be an arbitrary set of n complex

numbers.

Let t be a formal differential operator of order n on the interval I. Suppose that g is

a measurable

**complex**-**valued**function integrable over every compactsubinterval of I. Let to e I, and let co, c1, ..., c, 1 be an arbitrary set of n complex

numbers.

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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