## Linear Operators: Spectral operators |

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Page 1174

is in none of the sets e. Then (43) | g(s)

the similar well-known equation for scalarvalued functions. This concludes the ...

**Let**h be a scalar-valued function and put g(s) = h(s)a? if s is in e, and g(s) = 0 if sis in none of the sets e. Then (43) | g(s)

**f**(s)is = sh's)**f**(s) as: hence (42) follows fromthe similar well-known equation for scalarvalued functions. This concludes the ...

Page 1200

This operator f(T) is also the same as the operator x01 + . . . .x,T" as defined in

Definition 1.1. PRoof. It is clear that the sum zol-1-... +x,T" defined in accordance

with Definition 1.1 is the same as the operator f(T) of Definition VII.9.6.

...

This operator f(T) is also the same as the operator x01 + . . . .x,T" as defined in

Definition 1.1. PRoof. It is clear that the sum zol-1-... +x,T" defined in accordance

with Definition 1.1 is the same as the operator f(T) of Definition VII.9.6.

**Let f**(T) be...

Page 1649

Js or of rath. f so, i. to: f : If s to : o: Fo • *, 7 DEFINITION. Let I be an open set in E",

and

F(q) = F(j), p e Co(I), is called the compler conjugate of F. 8 LEMMA. Let I and F ...

Js or of rath. f so, i. to: f : If s to : o: Fo • *, 7 DEFINITION. Let I be an open set in E",

and

**let F**be a distribution in I. Then the distribution F in I defined by the equationF(q) = F(j), p e Co(I), is called the compler conjugate of F. 8 LEMMA. Let I and F ...

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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