## Linear Operators: Spectral theory |

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Page 906

A bounded operator T in Hilbert space ^) is called

called self adjoint, symmetric or Hermitian if T = T*; positive if it is self adjoint and

if (Tx, x) 2; 0 for every x in $); and positive definite if it is positive and (Tx, x) > 0 for

...

A bounded operator T in Hilbert space ^) is called

**unitary**if TT* = T*T — I; it iscalled self adjoint, symmetric or Hermitian if T = T*; positive if it is self adjoint and

if (Tx, x) 2; 0 for every x in $); and positive definite if it is positive and (Tx, x) > 0 for

...

Page 931

Applications are made to the restrictions of normal and

, Lumer, and Schaffer [1] proved that the restriction of a normal operator may

possess an inverse but not a square root (see also Halmos and Lumer [1]).

Dilations ...

Applications are made to the restrictions of normal and

**unitary**operators. Halmos, Lumer, and Schaffer [1] proved that the restriction of a normal operator may

possess an inverse but not a square root (see also Halmos and Lumer [1]).

Dilations ...

Page 1146

The following theorem is easily proved by induction in case R is

follows in the general case by the theorem stated above. Theorem. Any finite

dimensional representation of a compact group G is a direct sum of irreducible ...

The following theorem is easily proved by induction in case R is

**unitary**, and thusfollows in the general case by the theorem stated above. Theorem. Any finite

dimensional representation of a compact group G is a direct sum of irreducible ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

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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