## Linear Operators, Part 2 |

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Page 1187

The inverse of a

only if its domain is

which maps [12, y] into [y, ac] then I'(T-1) = A1I'(T) which shows that T is

and ...

The inverse of a

**closed**operator is**closed**. A bounded operator is**closed**if andonly if its domain is

**closed**. PROOF. If A1 is the isometric automorphism in Qwhich maps [12, y] into [y, ac] then I'(T-1) = A1I'(T) which shows that T is

**closed**ifand ...

Page 1436

By Corollary IV.3.2, every finite dimensional subspace of a. B-space is

Thus, by the HahnBanach theorem (II.3.l8) there exists a set ac:-', . . ., wt of

continuous linear functionals on the B-space such that .z':'(¢,) = 0 for 0 g i ¢ jg Ic, .

c:'(<p,) ...

By Corollary IV.3.2, every finite dimensional subspace of a. B-space is

**closed**.Thus, by the HahnBanach theorem (II.3.l8) there exists a set ac:-', . . ., wt of

continuous linear functionals on the B-space such that .z':'(¢,) = 0 for 0 g i ¢ jg Ic, .

c:'(<p,) ...

Page 1902

... finite dimensional space, V11.1.10 (560) in general space, VII.8.9 (568)

remarks on, (607-609), (612) for unbounded

Cauchy integral theorem, (225) Cauchy problem, (618-614), (689-641) Cauchy

sequence, ...

... finite dimensional space, V11.1.10 (560) in general space, VII.8.9 (568)

remarks on, (607-609), (612) for unbounded

**closed**operators, V11.9.4 (601)Cauchy integral theorem, (225) Cauchy problem, (618-614), (689-641) Cauchy

sequence, ...

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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