## Linear Operators: Self Adjoint Operators in Hilbert Space. Spectral theory. Part II |

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Page 890

One class of scalar

has already been defined is the class ... determined operator in Hilbert space

which corresponds to the continuous scalar

One class of scalar

**functions f**, other than polynomials, for which the operator /(T)has already been defined is the class ... determined operator in Hilbert space

which corresponds to the continuous scalar

**function f**under this "-isomorphism.Page 891

scalar

chapter we shall only integrate bounded

discussion of the integral will be restricted to that case. Let Z be a field of subsets

of a set ...

scalar

**function f**with respect to the operator valued set function E. In the presentchapter we shall only integrate bounded

**functions f**and so the followingdiscussion of the integral will be restricted to that case. Let Z be a field of subsets

of a set ...

Page 951

(a) If f is 1-measurable, then the

b) For f, geL1(R) the

the convolution f* g of f and g, which is defined by the equation </*g><w> ...

(a) If f is 1-measurable, then the

**function f**(a:—y) is a i.><1-measurable function. (b) For f, geL1(R) the

**function f**(.2:—y)g(y) is integrable in y for almost all m andthe convolution f* g of f and g, which is defined by the equation </*g><w> ...

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Preliminary Notions | 865 |

Copyright | |

61 other sections not shown

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additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity Nauk neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero