## Linear Operators: Spectral theory |

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Page 906

A bounded operator T in Hilbert space ^) is called unitary if TT* = T*T — I; it is

called self adjoint, symmetric or Hermitian if T = T*;

if (Tx, x) 2; 0 for every x in $); and

...

A bounded operator T in Hilbert space ^) 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 (Tx, x) 2; 0 for every x in $); and

**positive**definite if it is**positive**and (Tx, x) > 0 for...

Page 1247

Q.E.D. Next we shall require some information on

transformations and their square roots. 2 Lemma. A self adjoint transformation T

is

resolution of ...

Q.E.D. Next we shall require some information on

**positive**self adjointtransformations and their square roots. 2 Lemma. A self adjoint transformation T

is

**positive**if and only if a(T) is a subset of the interval [0, oo). Proof. Let E be theresolution of ...

Page 1338

Let be a

to a

equations = j m^fiidX), where e is any bounded Borel set, then the matrix {mtj(X)}

is ...

Let be a

**positive**matrix measure whose elements fiu are continuous with respectto a

**positive**a-finite measure fi. If the matrix of densities {m,,} is defined by theequations = j m^fiidX), where e is any bounded Borel set, then the matrix {mtj(X)}

is ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

Copyright | |

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### Common terms and phrases

Acad adjoint extension adjoint operator algebra Amer analytic B-algebra B*-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers constant 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 g Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping Math matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive 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