## Linear Operators: Spectral theory |

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Page 1297

If A(f) = 0 for each function in the domain of Tj(t) which vanishes in a

neighborhood of a, A will be

boundary value at b is defined similarly. By analogy with Definition XII. 4.25 an

equation B{f) = 0, ...

If A(f) = 0 for each function in the domain of Tj(t) which vanishes in a

neighborhood of a, A will be

**called**a boundary value at a. The concept of aboundary value at b is defined similarly. By analogy with Definition XII. 4.25 an

equation B{f) = 0, ...

Page 1432

In this case, v is

there is no singularity at all, and zero is

equation. If v = 1, the singularity of equation [*] at zero is

singularity; ...

In this case, v is

**called**the order of the singularity of equation [*] at zero. If v = 0,there is no singularity at all, and zero is

**called**a regular point of the differentialequation. If v = 1, the singularity of equation [*] at zero is

**called**a regularsingularity; ...

Page 1451

The number c is

the upper (lower) bound for T. If (Tx, x)~^0 for all x in %(T), then T is

negative. 20 Definition. Let r be a formally symmetric formal differential operator.

The number c is

**called**a bound for T, and the smallest (largest) such c is**called**the upper (lower) bound for T. If (Tx, x)~^0 for all x in %(T), then T is

**called**non-negative. 20 Definition. Let r be a formally symmetric formal differential operator.

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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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 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 q Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma Proc prove real axis real numbers representation satisfies Section sequence singular solution spectral spectral theory square-integrable subspace Suppose symmetric operator theory topology transform unique unitary vanishes vector zero