## Linear Operators: Spectral operators |

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Page 984

The set of functions f in L1(R) for which f vanishes in a neighborhood of infinity is

(R, 2, u) which vanish outside of compact sets is

The set of functions f in L1(R) for which f vanishes in a neighborhood of infinity is

**dense**in L1(R). PRoof. It follows from Lemma 3.6 that the set of all functions in L2(R, 2, u) which vanish outside of compact sets is

**dense**in this space, and from ...Page 1245

The Canonical Factorization In this section we shall prove that each closed

operator T with

where A is a positive (i.e., (Air, r) > 0, a e Q(A)) self adjoint transformation, and P

is a ...

The Canonical Factorization In this section we shall prove that each closed

operator T with

**dense**domain in Hilbert space has a unique factorization T = PA,where A is a positive (i.e., (Air, r) > 0, a e Q(A)) self adjoint transformation, and P

is a ...

Page 1905

... (233) Deficiency indices and spaces, definition, XII.4.9 (1226) De Morgan,

rules of, (2)

438–439)

and ...

... (233) Deficiency indices and spaces, definition, XII.4.9 (1226) De Morgan,

rules of, (2)

**Dense**convex sets, V.7.27 (437)**Dense**linear manifolds, V.7.40–41 (438–439)

**Dense**set, definition, I.6.11 (21) density of continuous functions in TMand ...

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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