## Linear Operators: Spectral theory |

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Page 984

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

L2(R, 3, 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 inL2(R, 3, u) which vanish outside of compact sets is

**dense**in this space, and ...Page 1188

If the domain Q(T) of the operator T is

by definition, of all y in S) for which (Tr, y) is continuous for a in 3)(T). Since 3)(T)

is

If the domain Q(T) of the operator T is

**dense**in S) then the domain £(T") consists,by definition, of all y in S) for which (Tr, y) is continuous for a in 3)(T). Since 3)(T)

is

**dense**in § there is (IV.4.5) a uniquely determined point y” in S) such that (Tr, ...Page 1271

frequently-used device, it is appropriate that we give a brief sketch indicating how

the Cayley transform can be used to determine when a symmetric operator has a

self adjoint extension. Let T be a symmetric operator with domain Q(T)

frequently-used device, it is appropriate that we give a brief sketch indicating how

the Cayley transform can be used to determine when a symmetric operator has a

self adjoint extension. Let T be a symmetric operator with domain Q(T)

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

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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