## Linear Operators, Part 2 |

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Page 1010

resentations as an L2-space and in this setting the class of

space A bounded linear operator T is said to be a

case the ...

resentations as an L2-space and in this setting the class of

**HilbertSchmidt****operators**may be defined as follows. ... a complete orthonormal set in the Hilbertspace A bounded linear operator T is said to be a

**Hilbert**-**Schmidt operator**incase the ...

Page 1013

The operator T is compact (cf. Exercise X.8.5), but it is not in HS. It has been

noted in the preceding discussion that the class of

forms a Banach algebra (without identity) under the norm ' . It may readily be

shown that ...

The operator T is compact (cf. Exercise X.8.5), but it is not in HS. It has been

noted in the preceding discussion that the class of

**Hilbert**-**Schmidt operators**forms a Banach algebra (without identity) under the norm ' . It may readily be

shown that ...

Page 1132

If K is a

kernels representing K in the sense that <s> Km)./.<8>.---1= Łg.<~»>.g.<s>. - - -1

where <4) gm = E f1K..-<8. 1>/,~<˘>dt Ii-1 0 the series converging

unconditionally in ...

If K is a

**Hilbert**-**Schmidt operator**in L,(A), there exists a unique set K,,(s, t) ofkernels representing K in the sense that <s> Km)./.<8>.---1= Łg.<~»>.g.<s>. - - -1

where <4) gm = E f1K..-<8. 1>/,~<˘>dt Ii-1 0 the series converging

unconditionally in ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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### Common terms and phrases

Acad adjoint extension adjoint operator algebra Amer analytic B-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients 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 hypothesis identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Paoor partial differential operator Pnoor positive preceding lemma Proc prove real axis real numbers representation satisfies second order Section sequence singular solution spectral set spectral theory square-integrable subspace Suppose symmetric operator topology transform unique unitary vanishes vector zero