Linear Operators: Spectral operators |
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Page 2144
... topology of X * , and is bounded . It remains only to show that A ( o ) A ( S ) = A ( od ) . It is seen , by using the above remarks , that for a fixed σ the family of S for which the equation is valid is a o - field . Thus if σ is in S ...
... topology of X * , and is bounded . It remains only to show that A ( o ) A ( S ) = A ( od ) . It is seen , by using the above remarks , that for a fixed σ the family of S for which the equation is valid is a o - field . Thus if σ is in S ...
Page 2203
... topology of A. Since σ ( E ) σ ( F ) = σ ( EF ) , the sets o ( E ) actually form a basis for the topology in A ( cf. Theorem IX.2.11 and its proof ) . Consider now a closed set d1 with d≤ 81e . Each point λ in 81 is interior to some ...
... topology of A. Since σ ( E ) σ ( F ) = σ ( EF ) , the sets o ( E ) actually form a basis for the topology in A ( cf. Theorem IX.2.11 and its proof ) . Consider now a closed set d1 with d≤ 81e . Each point λ in 81 is interior to some ...
Page 2319
... topology of An ) ( J ) . ∞ = PROOF . The series 1 E ( A ; T ) f certainly converges uncondi- tionally in the topology of L2 ( J ) . On the other hand , so does the series = T ( Σ E ( A ,; T ) S ) – Î ̧ B ( \ , ; T ) ( TS ) i = 1 ( cf ...
... topology of An ) ( J ) . ∞ = PROOF . The series 1 E ( A ; T ) f certainly converges uncondi- tionally in the topology of L2 ( J ) . On the other hand , so does the series = T ( Σ E ( A ,; T ) S ) – Î ̧ B ( \ , ; T ) ( TS ) i = 1 ( cf ...
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