## Linear Operators: Spectral theory |

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Page 984

The set of functions f in L^R) for which f vanishes in a neighborhood of infinity is

R, n) which vanish outside of compact sets is

The set of functions f in L^R) for which f vanishes in a neighborhood of infinity is

**dense**in L^R). Proof. It follows from Lemma 3.6 that the set of all functions in L2(R, n) which vanish outside of compact sets is

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

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

operator T with

where A is a positive (i.e., (Ax, x) ^ 0, xe^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., (Ax, x) ^ 0, xe^A)) self adjoint transformation, and P is a

...

Page 1246

We may also regard A as a mapping from the

space In this case A is still continuous, for \Ax\\ = {Ax, Ax\ = (A2x, x\ = {Ax, x), x e

and, by the inequalities above, (Ax, x) \Ax\ \x\ 5g I^vtIjIj-I, showing that \Ax^ ^ \x\.

We may also regard A as a mapping from the

**dense**subspace %(T) of into thespace In this case A is still continuous, for \Ax\\ = {Ax, Ax\ = (A2x, x\ = {Ax, x), x e

and, by the inequalities above, (Ax, x) \Ax\ \x\ 5g I^vtIjIj-I, showing that \Ax^ ^ \x\.

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

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Acad adjoint extension adjoint operator algebra Amer analytic B-algebra B*-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers constant 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 g Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping Math matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive 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