## Linear Operators, Part 2 |

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Page 1247

If T is a positive self adjoint transformation, there is a

transformation A such that A3 = T. Pnoor. By Lemma 2, o(T) Q [0, co) and, by

Theorem 2.6(d), the positive function f(/1) = 1* on a(T) defines the self adjoint ...

If T is a positive self adjoint transformation, there is a

**unique**positive self adjointtransformation A such that A3 = T. Pnoor. By Lemma 2, o(T) Q [0, co) and, by

Theorem 2.6(d), the positive function f(/1) = 1* on a(T) defines the self adjoint ...

Page 1250

Finally we show that the decomposition T = PA of the theorem is

Lemma l.6(c), AP' = T'. Hence T*T = AP'PA. Since, by Lemma 5, P*P is a

projection onto §li(A), it follows that T'T = A2. The uniqueness of A now follows

from Lemma 3.

Finally we show that the decomposition T = PA of the theorem is

**unique**. ByLemma l.6(c), AP' = T'. Hence T*T = AP'PA. Since, by Lemma 5, P*P is a

projection onto §li(A), it follows that T'T = A2. The uniqueness of A now follows

from Lemma 3.

Page 1513

Let Ft be the

and exponents which has the form z“*'(1 +z+ . . .) near z = 0. Then, since F, and F:

together comprise a basis for the solutions of our equation, we have a relation ...

Let Ft be the

**unique**solution of the equation with these same regular singularitiesand exponents which has the form z“*'(1 +z+ . . .) near z = 0. Then, since F, and F:

together comprise a basis for the solutions of our equation, we have a relation ...

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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