## Linear Operators: Spectral operators |

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Page 940

The function j is said to be

the proof of the following theorem and so the field of scalars is taken to be the

field ...

The function j is said to be

**finite dimensional**if its set {f's e G} of translates is a**finite dimensional**vector space of functions. The spectral theorem will be used inthe proof of the following theorem and so the field of scalars is taken to be the

field ...

Page 1092

Let & be a

of To; suppose that the dimension of & is d. Then, plainly, & is invariant under T

and T*, and, since (T&#, r) = (&#, Tow) = 0 for all a, we have T&# = 0 and similarly

...

Let & be a

**finite**-**dimensional**space including both the range of T and the rangeof To; suppose that the dimension of & is d. Then, plainly, & is invariant under T

and T*, and, since (T&#, r) = (&#, Tow) = 0 for all a, we have T&# = 0 and similarly

...

Page 1146

Any

irreducible representations. This theorem shows that in studying

generality, confine ...

Any

**finite dimensional**representation of a compact group G is a direct sum ofirreducible representations. This theorem shows that in studying

**finite****dimensional**representations of a compact group G we may, without loss ofgenerality, confine ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

52 other sections not shown

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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