## Linear Operators, Part 2 |

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Page 1211

Under

the Cartesian product of S and the real number system R which is measurable

with respect to the product of v and the measure /4 = (E(-)g, g), and which has the

...

Under

**Hypothesis**7, there is, for each g in L2(S, Z', v), a function W defined onthe Cartesian product of S and the real number system R which is measurable

with respect to the product of v and the measure /4 = (E(-)g, g), and which has the

...

Page 1215

Hence the expansion f = 2,,“ fa determined by the direct sum decomposition $3 =

Z.{7a takes the form stated in the following corollary. 12 COROLLARY. With the

notation and

Hence the expansion f = 2,,“ fa determined by the direct sum decomposition $3 =

Z.{7a takes the form stated in the following corollary. 12 COROLLARY. With the

notation and

**hypothesis**of the preceding theorem we have no =a§J:°(Uf)..(1)W..Page 1733

l('-7+KlA'l(/°sZl_/ills,-an = lA_l(E°SZ'*€ll<k,-2») is uniformly bounded in A. Since,

by the inductive

onto HI,:'l(C) n Hf,:l°(C), it follows that |A"1(foS-—/)](,,°) is uniformly bounded in A

...

l('-7+KlA'l(/°sZl_/ills,-an = lA_l(E°SZ'*€ll<k,-2») is uniformly bounded in A. Since,

by the inductive

**hypothesis**, (o+K)" is a uniformly bounded mapping of Hf,':'"")(C)onto HI,:'l(C) n Hf,:l°(C), it follows that |A"1(foS-—/)](,,°) is uniformly bounded in A

...

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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