## Linear Operators, Part 2 |

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Page 1015

If lim T” = T in the

the integral in [*] contains a(T,,) for all sufficiently large n. From Corollary VII.6.8 it

is seen that, in the

If lim T” = T in the

**norm**of HS it follows from Lemma VII.6.5 that the contour C ofthe integral in [*] contains a(T,,) for all sufficiently large n. From Corollary VII.6.8 it

is seen that, in the

**norm**of HS+, lim [1, —T,,]"1 = [1, —T]-1 uniformly for 1 in C.Page 1297

Now T,(-r) is an adjoint (Theorem 10); therefore (cf. XII.1.6) SD(T1('r)) is complete

in the

element of H "(J ) it follows easily that §D(T1('r)) is also complete under the

1f|,.

Now T,(-r) is an adjoint (Theorem 10); therefore (cf. XII.1.6) SD(T1('r)) is complete

in the

**norm**|f11. Since the two additional terms in [f|2 are the**norm**off as anelement of H "(J ) it follows easily that §D(T1('r)) is also complete under the

**norm**1f|,.

Page 1639

J J' ml This

If k < 00 and I is compact, but not otherwise, the spaces C"(I) and C:(I) are B-

spaces under a

J J' ml This

**norm**makes each of the spaces listed above into a complete F-space.If k < 00 and I is compact, but not otherwise, the spaces C"(I) and C:(I) are B-

spaces under a

**norm**equivalent to the**norm**displayed, though not under the**norm**...### What people are saying - Write a review

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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