## Linear Operators: Spectral theory |

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Page 1040

y^X) is

X; T)*y vanishes which will prove that y(X) is

that y(X) can only fail to be

y^X) is

**analytic**even at X = Xm. It will now be shown that y2(X) = XNE(X~m; T)*R(X; T)*y vanishes which will prove that y(X) is

**analytic**at all the points X = Xm, sothat y(X) can only fail to be

**analytic**at the point A = 0. To show this, note that ...Page 1102

The determinant det(Z -\-zTn) is an

if Tn operates in finite-dimensional space, and hence more generally if Tn has a

finite-dimensional range. Thus, since a bounded convergent sequence of ...

The determinant det(Z -\-zTn) is an

**analytic**(and even a polynomial) function of 2,if Tn operates in finite-dimensional space, and hence more generally if Tn has a

finite-dimensional range. Thus, since a bounded convergent sequence of ...

Page 1364

It follows easily that {^,Qj(A)} has an inverse {P,,(A)}

PtM>PfoW = AeG(A„), and I PM(A)9>,V(A) = 22 P„(A)gt(A)^Qt(A0) >-l i-1 Jr-l = 2 =

fctf). A6G(Ao). It is clear from the last formula that gf is

It follows easily that {^,Qj(A)} has an inverse {P,,(A)}

**analytic**for AeG(A„). Thus 2PtM>PfoW = AeG(A„), and I PM(A)9>,V(A) = 22 P„(A)gt(A)^Qt(A0) >-l i-1 Jr-l = 2 =

fctf). A6G(Ao). It is clear from the last formula that gf is

**analytic**in G. Q.E.D. 18 ...### What people are saying - Write a review

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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