## Linear Operators, Part 2 |

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Page 861

I) if and only if a: has an inverse in X and that when this inverse T?

= T,_1. Clearly if ac-1

1!l)Z] = 3/Z» (T?!/lz = TZIU/Z), and if a = Tgle, then az = Tfz for every z e I. Also .

I) if and only if a: has an inverse in X and that when this inverse T?

**exists**, then T?= T,_1. Clearly if ac-1

**exists**then T,-1T, = TQTT' =I. If T?**exists**in B(&'), then T=l(T;1!l)Z] = 3/Z» (T?!/lz = TZIU/Z), and if a = Tgle, then az = Tfz for every z e I. Also .

Page 1057

Thus (2) gives F<K =-=1><u> = <2»)-"/2 lim efgfl my) UEne""'f(w—z/>011} dy _ .

!?(y) ,.,,, provided only that the limit in the braces in this last equation

to complete the proof of the present lemma, it suffices to show that Q(y) . . Q(y) .

Thus (2) gives F<K =-=1><u> = <2»)-"/2 lim efgfl my) UEne""'f(w—z/>011} dy _ .

!?(y) ,.,,, provided only that the limit in the braces in this last equation

**exists**. Thus,to complete the proof of the present lemma, it suffices to show that Q(y) . . Q(y) .

Page 1262

Then there

such that Aa: = PQ.z', .2: e Q, P denoting the orthogonal projection of Si), on Q).

29 Let {Tn} be a sequence of bounded operators in Hilbert space .8} Then there ...

Then there

**exists**a Hilbert space 52), QQ, and an orthogonal projection Q in {)1such that Aa: = PQ.z', .2: e Q, P denoting the orthogonal projection of Si), on Q).

29 Let {Tn} be a sequence of bounded operators in Hilbert space .8} Then there ...

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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