## Linear Operators, Part 2 |

### From inside the book

Results 1-3 of 22

Page 917

where

elements of Let

that if F is a bounded Borel function and y is in then (UF(T)y),,(J.) = F(1)(Uy),,(1), ...

where

**fi**is in L2(e,, Ia), will be used for the elements of Q and a similar one for theelements of Let

**V**= UU'1, so that**V**is an isometric map off? onto 3,1'. We recallthat if F is a bounded Borel function and y is in then (UF(T)y),,(J.) = F(1)(Uy),,(1), ...

Page 1373

are inverse matrices, it follows readily that AB = BA = I. Thus, A and B are

isometric isomorphisms onto all of L,(/1, {

/e§>>(G<T>)It follows in the same way that all these statements hold if L2(/1, {

}) is ...

are inverse matrices, it follows readily that AB = BA = I. Thus, A and B are

isometric isomorphisms onto all of L,(/1, {

**fi**”}) ... <1)=G<1><**v**/>.<1>. 1'=1.....n. 16/1./e§>>(G<T>)It follows in the same way that all these statements hold if L2(/1, {

**fi**,-,}) is ...

Page 1377

Let)', and 11 be the end points of /1. Then (i) the inverse 0/ the isometric

isomorphism

lim JV' { 2

(/1, {

Let)', and 11 be the end points of /1. Then (i) the inverse 0/ the isometric

isomorphism

**V**of E(/1)L,(I) onto L2(/1, {**fi**,_,}) is given by the formula l x (**V**“F)(t) =lim JV' { 2

**Fi**(,l.)a,(t,}.),8,,(d}.)}, ,4 -A )4 1 1-1 U 0 0 '**Fi**"'\i where F = [F1, . . ., Fk] e L2(/1, {

**fi**,~ ...### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

13 other sections not shown

### Other editions - View all

### 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