## Linear Operators, Part 2 |

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Page 906

A bounded operator T in Hilbert space Q is called

called self adjoint, symmetric or Hermitian if T = T"'; positive if it is self adjoint and

if (Tm, 2:) g 0 for every .2: in Q3; and positive definite if it is positive and (Tm, w) >

0 ...

A bounded operator T in Hilbert space Q 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 (Tm, 2:) g 0 for every .2: in Q3; and positive definite if it is positive and (Tm, w) >

0 ...

Page 1146

Rn, and say that Ris the directsum of R1, . . ., Rn. The following theorem is easily

proved by induction in case R is

the theorem stated above. THEOREM. Any finite dimensional representation of a

...

Rn, and say that Ris the directsum of R1, . . ., Rn. The following theorem is easily

proved by induction in case R is

**unitary**, and thus follows in the general case bythe theorem stated above. THEOREM. Any finite dimensional representation of a

...

Page 1148

Moreover, H is the direct sum of a finite number of groups H,-, each of which is

either (1) The additive group of the real axis; or (2) The group SU(n) of all n><n

complex

Moreover, H is the direct sum of a finite number of groups H,-, each of which is

either (1) The additive group of the real axis; or (2) The group SU(n) of all n><n

complex

**unitary**matrices of determinant 1; or (3) The group SpU(n) of all 2nX2n ...### What people are saying - Write a review

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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### Common terms and phrases

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