Linear Operators, Part 2 |
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Page 1956
... THEOREM . If T is of finite type , its residual spectrum is void and a point is in its point spectrum if and only if ... follows from Theorem 2 that A is in the point spectrum of T. If E ( { λ } ) = 0 then it follows from Theorem 2 that ...
... THEOREM . If T is of finite type , its residual spectrum is void and a point is in its point spectrum if and only if ... follows from Theorem 2 that A is in the point spectrum of T. If E ( { λ } ) = 0 then it follows from Theorem 2 that ...
Page 2194
Nelson Dunford, Jacob T. Schwartz. PROOF . The first statement follows from Theorem 3. Let A1 be the algebra of all operators of the form ƒ ƒ ( \ ) E ( dλ ) where ƒ is E - essentially bounded on σ ( S ) . It follows from Theorem 10 that ...
Nelson Dunford, Jacob T. Schwartz. PROOF . The first statement follows from Theorem 3. Let A1 be the algebra of all operators of the form ƒ ƒ ( \ ) E ( dλ ) where ƒ is E - essentially bounded on σ ( S ) . It follows from Theorem 10 that ...
Page 2243
... follows from Theorem XV.5.1 that ƒ ( T | E ( en ) X ) = Sen ƒ ( ^ ) F ( dλ ) , so that ƒ ( T | E ( en ) X ) E ( en ) x = √ ƒ ( \ ) E ( d \ ) x , x Є X. It follows by Theorem 11 that en f ( T | E ( en ) X ) E ( en ) x = ƒ2 ( T ) E ( en ) ...
... follows from Theorem XV.5.1 that ƒ ( T | E ( en ) X ) = Sen ƒ ( ^ ) F ( dλ ) , so that ƒ ( T | E ( en ) X ) E ( en ) x = √ ƒ ( \ ) E ( d \ ) x , x Є X. It follows by Theorem 11 that en f ( T | E ( en ) X ) E ( en ) x = ƒ2 ( T ) E ( en ) ...
Contents
SPECTRAL OPERATORS | 1924 |
The Canonical Reduction of a Spectral Operator | 1939 |
Bounded Spectral Operators in Hilbert Space | 1947 |
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A₁ adjoint operator algebra of projections Amer analytic arbitrary B-algebra B*-algebra B₁ Banach space Boolean algebra Borel sets boundary conditions bounded Borel function bounded linear operator bounded operator closed operator commuting compact complex numbers complex plane converges Corollary countably additive Definition denote dense differential operator Doklady Akad domain eigenvalues elements equation exists finite number follows from Lemma formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis identity inequality integral invariant inverse L₁ Lebesgue Lemma Math multiplicity Nauk SSSR norm operators in Hilbert perturbation polynomial PROOF properties prove quasi-nilpotent resolution Russian S₁ satisfies scalar type operator scalar type spectral Section sequence shows spectral measure spectral operator spectral theory spectrum subset subspace Suppose trace class type spectral operator unbounded uniformly bounded unique vector zero