## Linear Operators: Spectral operators |

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Page 996

of g, and let St be the conjugate-orthogonal complement in L (R) of S. Then St is

closed under translation. Moreover, the set of characters in St consists precisely ...

**Let Q**be the closed linear subspace of L1(R) which is spanned by the translatesof g, and let St be the conjugate-orthogonal complement in L (R) of S. Then St is

closed under translation. Moreover, the set of characters in St consists precisely ...

Page 997

in the Li-closure of the linear manifold in L. (R) spanned by these characters. It

follows from Corollary W.3.12 that if |f(x)|r, mode – 0. i = 1,..., r, for some function f

in ...

**Let q**(q) consist of the characters [", ml], ..., [·, m, 1. According to Theorem 20, q isin the Li-closure of the linear manifold in L. (R) spanned by these characters. It

follows from Corollary W.3.12 that if |f(x)|r, mode – 0. i = 1,..., r, for some function f

in ...

Page 1437

Then, since To(r) C Ti(r), it follows immediately from the preceding lemma that

Zoe G.(Ti(r)), so that by Definition 6.1, Žo e o (t). Conversely,

be the closure in the Hilbert space

restriction ...

Then, since To(r) C Ti(r), it follows immediately from the preceding lemma that

Zoe G.(Ti(r)), so that by Definition 6.1, Žo e o (t). Conversely,

**let**Zoe G.(r).**Let**},be the closure in the Hilbert space

**Q**(Ti(r)) of £(To(r)), and**let**To(t) be therestriction ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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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