## Linear Operators: Spectral operators |

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Page 974

—oo —co Now the expression s. e-(++++)"/2 dr -oxo defines an entire function of

z which is equal to s' e-oed, = (2-1)} for all real z and therefore for all z. Hence we

conclude that (rf)(m) = (2+); e-("")}**. By

—oo —co Now the expression s. e-(++++)"/2 dr -oxo defines an entire function of

z which is equal to s' e-oed, = (2-1)} for all real z and therefore for all z. Hence we

conclude that (rf)(m) = (2+); e-("")}**. By

**Plancherel's theorem**s' e-o-'do-J., ...Page 1155

orthonormal set in L2(R), which fact was also proved in Theorem 1.6. We leave it

to the reader to show that if R is the compact Abelian group of the real numbers

modulo ...

**Plancherel's theorem**asserts that the set of characters forms a completeorthonormal set in L2(R), which fact was also proved in Theorem 1.6. We leave it

to the reader to show that if R is the compact Abelian group of the real numbers

modulo ...

Page 1160

measure in R is regular, the set X of functions in Li o La(R) which vanish outside

of compact sets in R are dense in L2(R); by

in L.(R) and hence {rf - rgf, ge.Y) is dense in L(R). Now let f, g be in X and ...

measure in R is regular, the set X of functions in Li o La(R) which vanish outside

of compact sets in R are dense in L2(R); by

**Plancherel's theorem**(tffe 3 } is densein L.(R) and hence {rf - rgf, ge.Y) is dense in L(R). Now let f, g be in X and ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

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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