## Linear Operators: Spectral theory |

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Page 1094

Then by Minkowski's

ir,!,. In the same way ( 2 h-^Wps l^+l^l,. ln=0 J Thus the first assertion of (b)

follows on addition. The assertion of (b), for the range 0 < p ^ 1, follows in just the

same ...

Then by Minkowski's

**inequality**, (00 \1/J> / 00 U/P / 00 n=0 / U-0 J ln=0 J =s inip+ir,!,. In the same way ( 2 h-^Wps l^+l^l,. ln=0 J Thus the first assertion of (b)

follows on addition. The assertion of (b), for the range 0 < p ^ 1, follows in just the

same ...

Page 1105

We now pause to sharpen another of the

the paragraph following Lemma 9, the continuity of the norm function which

follows from the triangle

that ...

We now pause to sharpen another of the

**inequalities**of Lemma 9. ... was noted inthe paragraph following Lemma 9, the continuity of the norm function which

follows from the triangle

**inequality**of Lemma 14(d), and by Lemma 11, it followsthat ...

Page 1774

The above

follows from the postulates for § that the Schwarz

is zero. Hence suppose that x 0 ^ y. For an arbitrary complex number a 0 ^ (x+ay,

...

The above

**inequality**, known as the Schwarz**inequality**, will be proved first. Itfollows from the postulates for § that the Schwarz

**inequality**is valid if either x or yis zero. Hence suppose that x 0 ^ y. For an arbitrary complex number a 0 ^ (x+ay,

...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

Copyright | |

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### Common terms and phrases

Acad adjoint extension adjoint operator algebra Amer analytic B-algebra B*-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers constant 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 g Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping Math matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive 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