## Linear Operators: Spectral operators |

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Page 906

A bounded operator T in Hilbert space S) is called unitary if TTo = To T = I; it is

called self adjoint, symmetric or Hermitian if T = To;

if (Tr, r) > 0 for every a in \); and

A bounded operator T in Hilbert space S) is called unitary if TTo = To T = I; it is

called self adjoint, symmetric or Hermitian if T = To;

**positive**if it is self adjoint andif (Tr, r) > 0 for every a in \); and

**positive**definite if it is**positive**and (Tr, r) > 0 for ...Page 1247

Q.E.D. Next we shall require some information on

transformations and their square roots. 2 LEMMA. A self adjoint transformation T

is

resolution ...

Q.E.D. Next we shall require some information on

**positive**self adjointtransformations and their square roots. 2 LEMMA. A self adjoint transformation T

is

**positive**if and only if o(T) is a subset of the interval [0, co). PRoof. Let E be theresolution ...

Page 1338

Let {u} be a

respect to a

the equations p,(e)=|n,(A)p(dA), where e is any bounded Borel set, then the

matria.

Let {u} be a

**positive**matria measure whose elements u, are continuous withrespect to a

**positive**g-finite measure u. If the matria of densities {m,} is defined bythe equations p,(e)=|n,(A)p(dA), where e is any bounded Borel set, then the

matria.

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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### Common terms and phrases

adjoint extension adjoint operator algebra analytic B-algebra B-space Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients compact operator complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality infinity integral interval kernel Lemma Let f Let Q linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real numbers satisfies sequence singular ſº solution spectral spectral theorem square-integrable subspace Suppose symmetric operator theory To(r To(t topology transform uniformly unique unitary vanishes vector zero