## 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 |

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