## Linear Operators, Part 2 |

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Page 905

If the spectrum of the bounded normal operator T in Q) is countable then there is

an

Z80». 1/)2/. w e it and, for each .2, all but a countable number of the coefficients ...

If the spectrum of the bounded normal operator T in Q) is countable then there is

an

**orthonormal**basis B for $3 consisting of eigenvectors of T. Furthermore, w =Z80». 1/)2/. w e it and, for each .2, all but a countable number of the coefficients ...

Page 1010

Let {xv oz e A} be a complete

linear operator T is said to be a Hilbert-Schmidt operator in case the quantity

defined by the equation IITII = {Z lTw.l'}* aeA is finite. The number is sometimes

called ...

Let {xv oz e A} be a complete

**orthonormal**set in the Hilbert space A boundedlinear operator T is said to be a Hilbert-Schmidt operator in case the quantity

defined by the equation IITII = {Z lTw.l'}* aeA is finite. The number is sometimes

called ...

Page 1028

1 _ _ 1 rm /(ET) =;_Icj(}.)(1I—ET) 1d}. _ 41].: (I—E) + El /(7-)(U—T)“ d/1 = /(0)(I—E

)+E/(T) = Ef(T)2m C In much the same way it may be proved that f(T)E = ](TE),

which, since T = TE, shows that f(T)E = f(T). Let {.c“, at e A} be an

basis ...

1 _ _ 1 rm /(ET) =;_Icj(}.)(1I—ET) 1d}. _ 41].: (I—E) + El /(7-)(U—T)“ d/1 = /(0)(I—E

)+E/(T) = Ef(T)2m C In much the same way it may be proved that f(T)E = ](TE),

which, since T = TE, shows that f(T)E = f(T). Let {.c“, at e A} be an

**orthonormal**basis ...

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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