Linear Operators: Spectral operators |
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Page 2183
... clearly a subalgebra of A ( 7 ) . Let B denote its closure in the uniform topology of operators . By Theorem XV.4.5 and the fact that a scalar type operator is clearly in the uniformly closed algebra generated by the projections in its ...
... clearly a subalgebra of A ( 7 ) . Let B denote its closure in the uniform topology of operators . By Theorem XV.4.5 and the fact that a scalar type operator is clearly in the uniformly closed algebra generated by the projections in its ...
Page 2211
... clearly A1 € A , A ̧ = E , √ h2 ( \ ) E ( d \ ) , A A , x = Ax , A1E2 = A2 , | A , | ≤ K | A | , where K is a bound for B. Since A , is in A , there is a uniquely defined continuous function f , in C ( 4 ) with A , = T ( f ) . This ...
... clearly A1 € A , A ̧ = E , √ h2 ( \ ) E ( d \ ) , A A , x = Ax , A1E2 = A2 , | A , | ≤ K | A | , where K is a bound for B. Since A , is in A , there is a uniquely defined continuous function f , in C ( 4 ) with A , = T ( f ) . This ...
Page 2332
... clearly sufficient for us to show that for each ƒ in L2 ( 0 , ∞ ) , the sequence bm ( f ) defined for m ≥1 by the formula . ∞ bm ( f ) = e - mt f ( t ) dt is in l . We may clearly suppose without loss of generality that ƒ is positive ...
... clearly sufficient for us to show that for each ƒ in L2 ( 0 , ∞ ) , the sequence bm ( f ) defined for m ≥1 by the formula . ∞ bm ( f ) = e - mt f ( t ) dt is in l . We may clearly suppose without loss of generality that ƒ is positive ...
Contents
SPECTRAL OPERATORS | 1924 |
14 | 1983 |
Sufficient Conditions | 2134 |
Copyright | |
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A₁ adjoint operator algebra of projections Amer analytic arbitrary asymptotic B₁ Banach space Boolean algebra Borel set boundary conditions bounded Borel function bounded linear operator bounded operator commuting compact complete Boolean algebra complex numbers complex plane continuous functions converges Corollary countably additive Definition denote differential operator Doklady Akad domain E₁ eigenvalues elements equation exists finite number follows from Lemma follows from Theorem follows immediately formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis identity inequality inverse L₁ Lebesgue Math multiplicity Nauk SSSR norm operators in Hilbert perturbation PROOF properties prove quasi-nilpotent resolution Russian satisfies scalar type operator scalar type spectral Section sequence shows spectral measure spectral operator spectral theory spectrum subset Suppose trace class type spectral operator unbounded uniformly bounded vector zero