## Linear Operators, Part 2 |

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Page 1179

sends a scalar-valued function with the Fourier transform /(E) into the vector-

valued function whose nth component has the Fourier transform f,,(E) defined by (

65) ...

**PROOF**. We saw in the course of proving Theorem 25 that the mapping If whichsends a scalar-valued function with the Fourier transform /(E) into the vector-

valued function whose nth component has the Fourier transform f,,(E) defined by (

65) ...

Page 1459

present corollary follows from Corollary 7 and Definition 25(b). Q.E.D. 81

COROLLARY. Suppose in addition to the hypotheses of Theorem 8 that the

coefficients pk are ...

**PROOF**. It is obvious from Definition 20 that -r is bounded below. Thus thepresent corollary follows from Corollary 7 and Definition 25(b). Q.E.D. 81

COROLLARY. Suppose in addition to the hypotheses of Theorem 8 that the

coefficients pk are ...

Page 1750

\'\'e shall see, however, that this fact is needed in the course of the

Theorem 1, and shall prove it by a direct method where it is needed. Remark 2.

The theorem is false if no boundedness restriction is imposed on the coefficient ...

\'\'e shall see, however, that this fact is needed in the course of the

**proof**ofTheorem 1, and shall prove it by a direct method where it is needed. Remark 2.

The theorem is false if no boundedness restriction is imposed on the coefficient ...

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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