## Linear Operators, Part 2 |

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Page 1233

Then (T1—}.0I)-1 = R(}.o) is an

not more than |.f(1,,)]_1. Consequently, the series [-1 §0<1—1..>"R<A.>"+1

converges if 11-1,,| < |J(2,,)|. Since T, is closed, we have (T.-M) (/1-11))" R<1..>"+

1y ...

Then (T1—}.0I)-1 = R(}.o) is an

**everywhere**defined, bounded operator of normnot more than |.f(1,,)]_1. Consequently, the series [-1 §0<1—1..>"R<A.>"+1

converges if 11-1,,| < |J(2,,)|. Since T, is closed, we have (T.-M) (/1-11))" R<1..>"+

1y ...

Page 1402

2) are void, then /1a,('r) is void, and it follows from Theorem 6.13 that W1(', 1) G

L2(a, b) Ia,--almost

with evident slight modifications to show that if B(f) = 0 is a boundary condition ...

2) are void, then /1a,('r) is void, and it follows from Theorem 6.13 that W1(', 1) G

L2(a, b) Ia,--almost

**everywhere**in /1. The proof of Theorem 5.4- will then applywith evident slight modifications to show that if B(f) = 0 is a boundary condition ...

Page 1899

(See Field of sets) Almost

additive scalar set functions, Il1.1.11 (100) definition for vector-valued set

functions, 1V.10.6 (822) Almost periodic functions, definition, lV.2.25 (242) space

of, ...

(See Field of sets) Almost

**everywhere**(or /1-almost**everywhere**) definition foradditive scalar set functions, Il1.1.11 (100) definition for vector-valued set

functions, 1V.10.6 (822) Almost periodic functions, definition, lV.2.25 (242) space

of, ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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Acad adjoint extension adjoint operator algebra Amer analytic B-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers continuous function converges Corollary deficiency indices Definition denote dense differential equations Doklady Akad domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Paoor partial differential operator Pnoor positive preceding lemma Proc prove real axis real numbers representation satisfies second order Section sequence singular solution spectral set spectral theory square-integrable subspace Suppose symmetric operator topology transform unique unitary vanishes vector zero