## Linear Operators: Spectral theory |

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Page 1187

The inverse of a

only if its domain is

which maps sa, y] into [y, r) then I'(T-1) = A11'(T) which shows that T is

The inverse of a

**closed**operator is**closed**. A bounded operator is**closed**if andonly if its domain is

**closed**. PRoof. If A1 is the isometric automorphism in S) GS S)which maps sa, y] into [y, r) then I'(T-1) = A11'(T) which shows that T is

**closed**if ...Page 1393

Let T be a

such that the range of AI —T is not

and is denoted by o,(T). It is clear that o,(T) Co.(T). If t is a formal differential

operator ...

Let T be a

**closed**operator in Hilbert space. Then the set of complex numbers Asuch that the range of AI —T is not

**closed**is called the essential spectrum of Tand is denoted by o,(T). It is clear that o,(T) Co.(T). If t is a formal differential

operator ...

Page 1394

by what has been shown above )+9t, , is

establish the converse part of the present lemma under the additional hypothesis

that JR is one-dimensional, i.e., that Jo = {x r}. If Tre T?), then T()+9°) = T())), ...

by what has been shown above )+9t, , is

**closed**, it is sufficient for this purpose toestablish the converse part of the present lemma under the additional hypothesis

that JR is one-dimensional, i.e., that Jo = {x r}. If Tre T?), then T()+9°) = T())), ...

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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