Linear Operators, Part 2 |
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Page 1934
... seen that x * x ( § ) = 0 for all έ and all x * = x * . Hence , by Corollary II.3.14 , x ( § ) = 0 and thus x = ( ¿ I − T ) x ( § ) = 0. Q.E.D. → 4 THEOREM . Let T be a bounded spectral operator with resolution of the identity E , and ...
... seen that x * x ( § ) = 0 for all έ and all x * = x * . Hence , by Corollary II.3.14 , x ( § ) = 0 and thus x = ( ¿ I − T ) x ( § ) = 0. Q.E.D. → 4 THEOREM . Let T be a bounded spectral operator with resolution of the identity E , and ...
Page 2163
... seen from Corollary XV.3.7 that F ( § ) also commutes with the projec- tions in the range of E , that is , ( iii ) F ( ¿ ) E ( 0 ) = E ( 0 ) F ( § ) , ξερ ( Τ ) , for every Borel set σ . If = { 01 , ... , σn } , π ' = { σ1 ' , ... , σ'n ...
... seen from Corollary XV.3.7 that F ( § ) also commutes with the projec- tions in the range of E , that is , ( iii ) F ( ¿ ) E ( 0 ) = E ( 0 ) F ( § ) , ξερ ( Τ ) , for every Borel set σ . If = { 01 , ... , σn } , π ' = { σ1 ' , ... , σ'n ...
Page 2183
... seen that PA ( 7 ) = A ( B ) . Moreover , it is clear that ( I — P ) B = R , and since R is closed , it will follow ... seen that E1 ( m ) = 1 , E , ( m ) = 0 for i ‡ j . Hence U ( m ) = S ( m ) + R ( m ) = S ( m ) = x¡ ; thus sup | a ...
... seen that PA ( 7 ) = A ( B ) . Moreover , it is clear that ( I — P ) B = R , and since R is closed , it will follow ... seen that E1 ( m ) = 1 , E , ( m ) = 0 for i ‡ j . Hence U ( m ) = S ( m ) + R ( m ) = S ( m ) = x¡ ; thus sup | a ...
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