## Linear Operators: Spectral theory |

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Page 1250

Finally we show that the decomposition T = PA of the theorem is

Lemma 1.6(c), AP* = T*. Hence T*T = AP*PA. Since, by Lemma 5, P*P is a

projection onto 1R(A), it follows that T*T = A2. The uniqueness of A now follows

from ...

Finally we show that the decomposition T = PA of the theorem is

**unique**. ByLemma 1.6(c), AP* = T*. Hence T*T = AP*PA. Since, by Lemma 5, P*P is a

projection onto 1R(A), it follows that T*T = A2. The uniqueness of A now follows

from ...

Page 1283

Thus, equation (e') has the

(-lY&H. Since all the terms in equation (e) but the first are absolutely continuous,

it follows that F is absolutely continuous. Thus Theorem 1 is proved for the ...

Thus, equation (e') has the

**unique**solution (cf. Lemma VII.3.4) 00 F = (J+0)-*ff = 2(-lY&H. Since all the terms in equation (e) but the first are absolutely continuous,

it follows that F is absolutely continuous. Thus Theorem 1 is proved for the ...

Page 1383

With boundary conditions A and C, the

boundary condition t3ct = Act is sin y/Jt. With boundary conditions A, the

eigenvalues are consequently to be determined from the equation sin y/X = 0.

With boundary conditions A and C, the

**unique**solution of r3a = Act satisfying theboundary condition t3ct = Act is sin y/Jt. With boundary conditions A, the

eigenvalues are consequently to be determined from the equation sin y/X = 0.

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

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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