## Linear Operators: Spectral theory |

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Page 973

t mapping

e) = A(t(e)), e e B, where 3 is the family of Borel sets

measure

t mapping

**R**onto all**of R**with the properties [**r**, m) = e^*(*), a e**R**, m e**R**and 2:tu(e) = A(t(e)), e e B, where 3 is the family of Borel sets

**in R**and where 2 is Haarmeasure

**on R**. PRoof. For a fixed m in .40 =**R**, the character sa, m), a e**R**, is a ...Page 1291

Observe that

I) and that

case t is formally self adjoint it will follow that

Observe that

**To(r**) and Ti(t) are both unbounded densely defined operators in L2(I) and that

**To(r**) C Ti(t). Our next task is to prove that if t is regular Ti(t) = To(t”)*. Incase t is formally self adjoint it will follow that

**To(r**) C Ti(t) =**To(r**)*, showing that ...Page 1437

Suppose that a bounded sequence {fi} of elements of 3 (

–70)f,} converges but the sequence {f,} has no convergent subsequence. Then,

since

Suppose that a bounded sequence {fi} of elements of 3 (

**To(r**)) exists such that {(t–70)f,} converges but the sequence {f,} has no convergent subsequence. Then,

since

**To(r**) C Ti(t), it follows immediately from the preceding lemma that Zoe g ...### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

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adjoint extension adjoint operator algebra Amer analytic B-algebra Banach Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients complete 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 identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping Math matrix measure Nauk SSSR N.S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Plancherel's theorem positive Proc PRoof prove real numbers satisfies sequence singular ſº solution spectral spectral set spectral theory square-integrable subspace Suppose theory To(r topology transform unique unitary vanishes vector zero