## Linear Operators, Part 2 |

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Page 1207

It is clear from Zorn's lemma that there is a maximal set A in .17 for which the

spaces Q“, a e A, are

that no .c ab 0 is

It is clear from Zorn's lemma that there is a maximal set A in .17 for which the

spaces Q“, a e A, are

**orthogonal**. Thus to prove the lemma it suffices to observethat no .c ab 0 is

**orthogonal**to each of the spaces Q)“. Indeed, if .1: ;é 0 is**orthogonal**...Page 1227

(a) SD(T), $+, and $_ are closed

. (b) sou'-) = €D(T) ea 1*), ea §>_. Paoor. By Lemma 8(a), 5D(T) is closed.

Suppose {.z',,} is a sequence of elements of 553+ converging to 1' e 5D(T"), then

{[iz',,, ...

(a) SD(T), $+, and $_ are closed

**orthogonal**subspaces of the Hilbert space 'D(T"). (b) sou'-) = €D(T) ea 1*), ea §>_. Paoor. By Lemma 8(a), 5D(T) is closed.

Suppose {.z',,} is a sequence of elements of 553+ converging to 1' e 5D(T"), then

{[iz',,, ...

Page 1262

P denoting the

imaginary parts separately and use Exercise 27). 80 Let {An} be a sequence of

bounded self adjoint transformations in Hilbert space Suppose that there exists a

constant ...

P denoting the

**orthogonal**projection of 6;), onto (Hint: Consider real andimaginary parts separately and use Exercise 27). 80 Let {An} be a sequence of

bounded self adjoint transformations in Hilbert space Suppose that there exists a

constant ...

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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