## Linear Operators: Spectral operators |

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Page 1187

A bounded operator is

isometric automorphism in § G \\ which maps sa, y] into sy, a then T(T-1) = A11'(T)

which shows that T is

A bounded operator is

**closed**if and only if its domain is**closed**. PRoof. If A, is theisometric automorphism in § G \\ which maps sa, y] into sy, a then T(T-1) = A11'(T)

which shows that T is

**closed**if and only if T-1 is**closed**. If B is a bounded ...Page 1226

Q.E.D. It follows from Lemma 6(b) that any symmetric operator with dense domain

has a unique minimal

following definition. 7 DEFINITION. The minimal

Q.E.D. It follows from Lemma 6(b) that any symmetric operator with dense domain

has a unique minimal

**closed**symmetric extension. This fact leads us to make thefollowing definition. 7 DEFINITION. The minimal

**closed**symmetric extension of ...Page 1393

Then the set of complex numbers A such that the range of AI —T is not

called the essential spectrum of T and is denoted by o,(T). It is clear that a. (T) Co.

(T). If t is a formal differential operator defined on the interval I, then the essential

...

Then the set of complex numbers A such that the range of AI —T is not

**closed**iscalled the essential spectrum of T and is denoted by o,(T). It is clear that a. (T) Co.

(T). If t is a formal differential operator defined on the interval I, then the essential

...

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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