## Linear Operators: Spectral theory |

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Page 968

Let m , be an

neighborhood of hm .. By IV.8.19 the integrable continuous functions on R are

dense in L ( R ) so there is a continuous function f on R such that i < 1 and ( if ) (

mo ) + 0 ...

Let m , be an

**arbitrary**point in Mo , 0 << < 1 , and let N ( hem . , K , € ) , be aneighborhood of hm .. By IV.8.19 the integrable continuous functions on R are

dense in L ( R ) so there is a continuous function f on R such that i < 1 and ( if ) (

mo ) + 0 ...

Page 1179

Theorem 25 remains valid in the full range 1 < p < oo , and for functions f with

values in an

with trivial modifications of its proof to functions with values in an

...

Theorem 25 remains valid in the full range 1 < p < oo , and for functions f with

values in an

**arbitrary**Hilbert space . PROOF . Note that Theorem 20 goes overwith trivial modifications of its proof to functions with values in an

**arbitrary**Hilbert...

Page 1795

The Dirichlet problem for linear elliptic equations of

variable coefficients . Proc . Nat . Acad . Sci . U.S.A. 38 , 230-235 ( 1952 ) . 2 . The

Dirichlet and vibration problems for linear elliptic differential equations of

...

The Dirichlet problem for linear elliptic equations of

**arbitrary**even order withvariable coefficients . Proc . Nat . Acad . Sci . U.S.A. 38 , 230-235 ( 1952 ) . 2 . The

Dirichlet and vibration problems for linear elliptic differential equations of

**arbitrary**...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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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 neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result Russian satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero