## Linear operators: Spectral operators |

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Page 1937

E(A() which is equivalent to a diagonal

T — A, I)E(Xt). Stated in other terms, this classical reduction of Jordan asserts that

every finite square

E(A() which is equivalent to a diagonal

**matrix**and a nilpotent operator N = £?=i (T — A, I)E(Xt). Stated in other terms, this classical reduction of Jordan asserts that

every finite square

**matrix**of complex numbers is equivalent to the sum of a ...Page 2011

The notation will be that of the preceding section, but we shall now be concerned

with p x p

) we define the

The notation will be that of the preceding section, but we shall now be concerned

with p x p

**matrices**A(s) = (djk(s)) ... For every set a in E and every such**matrix**A(s) we define the

**matrix**(1) Aa(s) = A(s), sea, = 0, S$ar, and the operator A„ in ...Page 2327

are those of the inverse

(fx) depends analytically on p, 1 ^ i, k ^ n. Consider the element J?(A)/ = G(A)/-Jf(

M(A))-1 f Mlk^(X))(BiG(X)f)ak(fji(X)) of HW(I). Since (t - X)ok(n(X)) = 0, we have (A

...

are those of the inverse

**matrix**of the**matrix**defined by the equation (5). Then Mik(fx) depends analytically on p, 1 ^ i, k ^ n. Consider the element J?(A)/ = G(A)/-Jf(

M(A))-1 f Mlk^(X))(BiG(X)f)ak(fji(X)) of HW(I). Since (t - X)ok(n(X)) = 0, we have (A

...

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

47 other sections not shown

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adjoint operator algebra of projections Amer analytic arbitrary asymptotic B-space Banach space Boolean algebra Borel sets boundary conditions bounded Borel function bounded linear operator bounded operator commuting compact complex numbers complex plane constant contains continuous functions converges Corollary countably additive Definition denote dense differential operator disjoint Doklady Akad domain eigenvalues elements equation exists finite number Foias follows from Lemma follows from Theorem formal differential operator formula Hence Hilbert space hypothesis identity inequality inverse Lebesgue Math matrix measurable functions multiplicity Nauk SSSR norm normal operators operators in Hilbert perturbation polynomial Proof properties prove quasi-nilpotent restriction Russian satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum strong operator topology subset Suppose trace class type spectral operator unbounded uniformly bounded unique vector weakly complete zero