## Linear Operators, Part 2 |

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Page 906

A bounded operator T in Hilbert space Q is called unitary if TT* = T"'T = I; it is

called self adjoint, symmetric or Hermitian if T = T"';

if (Tm, 2:) g 0 for every .2: in Q3; and

0 ...

A bounded operator T in Hilbert space Q is called unitary if TT* = T"'T = I; it is

called self adjoint, symmetric or Hermitian if T = T"';

**positive**if it is self adjoint andif (Tm, 2:) g 0 for every .2: in Q3; and

**positive**definite if it is**positive**and (Tm, w) >0 ...

Page 1247

By Theorem 2.9(b) we have [=i=] a(T) u {0} 2 Uo(T,,) 2 tr(T). n=1 Thus, if a(T) Q [0,

oo), it follows from Theorem X.4-.2 that Tn g 0. Hence (Tm, .23) = lim" (Tam, av) g

0 and T is

By Theorem 2.9(b) we have [=i=] a(T) u {0} 2 Uo(T,,) 2 tr(T). n=1 Thus, if a(T) Q [0,

oo), it follows from Theorem X.4-.2 that Tn g 0. Hence (Tm, .23) = lim" (Tam, av) g

0 and T is

**positive**. Conversely, if T 2 0, then (T,,.z', .2') = (TE([—n, n]).z', E([—n, ...Page 1338

Let {,u,.,} be a

respect to a

by the equations #.,<e> = f;».,<1>a<d1>. where e is any bounded Borel set, then

...

Let {,u,.,} be a

**positive**matria: measure whose elements Ia” are continuous withrespect to a

**positive**or-finite measure /4. If the matrix of densities {mu} is definedby the equations #.,<e> = f;».,<1>a<d1>. where e is any bounded Borel set, then

...

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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