## Linear Operators, Part 2 |

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Page 922

, T2, . . ., are defined. 1 LEMMA. Let S, T, S", Tn, n g 1 be bounded linear

operators in Hilbert space with Sn —> S, Tn —> T in the strong operator

**topology**, i.e., Tam -> Ta: for every .2: in the space upon which the operators T, T1, T2, . . ., are defined. 1 LEMMA. Let S, T, S", Tn, n g 1 be bounded linear

operators in Hilbert space with Sn —> S, Tn —> T in the strong operator

**topology**.Page 1420

(a') The

sequence in §D(T1(r)). Suppose that {fa} converges to zero in the

T1(r)).

(a') The

**topology**of the Hilbert space 'D(T1(-r')) is the same as its relative**topology**as a subspace of the Hilbert space 'D(T1('r+"r')). Indeed, let {fn} be asequence in §D(T1(r)). Suppose that {fa} converges to zero in the

**topology**of $(T1(r)).

Page 1921

F (1550) Subadditive function, definition, (618) Subbase for a

criterion for, 1.4.8 (11) Subspace, of a linear space, (86). (See also Manifold)

Summability, of Fourier series, IV.14.84—51 (861-804) general principle of, Xl11.

9.

F (1550) Subadditive function, definition, (618) Subbase for a

**topology**, 1.4.0 (10)criterion for, 1.4.8 (11) Subspace, of a linear space, (86). (See also Manifold)

Summability, of Fourier series, IV.14.84—51 (861-804) general principle of, Xl11.

9.

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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