## Linear Operators: Spectral theory |

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Page 1010

Let {r, o, e A} be a complete

linear operator T is said to be a Hilbert-Schmidt operator in case the quantity |T|

defined by the equation T-X, to is finite. The number |T| is sometimes called the ...

Let {r, o, e A} be a complete

**orthonormal**set in the Hilbert space Sy. A boundedlinear operator T is said to be a Hilbert-Schmidt operator in case the quantity |T|

defined by the equation T-X, to is finite. The number |T| is sometimes called the ...

Page 1028

In much the same way it may be proved that f(T)F = f(TE), which, since T = TE,

shows that f(T) E = f(T). Let {r., & e A} be an

finite dimensional we may suppose without loss of generality that there is a finite

...

In much the same way it may be proved that f(T)F = f(TE), which, since T = TE,

shows that f(T) E = f(T). Let {r., & e A} be an

**orthonormal**basis for $5. Since ES) isfinite dimensional we may suppose without loss of generality that there is a finite

...

Page 1779

A set A is called an

Every closed linear manifold in S) contains an

PROOF.

A set A is called an

**orthonormal**basis for the linear manifold 92 in S) if A is an**orthonormal**set contained in J. and if a = X (r, y)), a e )?. we A 12 THEOREM.Every closed linear manifold in S) contains an

**orthonormal**basis for itself.PROOF.

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

48 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero