Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |
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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 let ...
... 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 let ...
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 ) , ... 9 ... " for every Borel set σ . If π = { 01 , On } , π ' = { 01 ' , o ...
... 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 ) , ... 9 ... " for every Borel set σ . If π = { 01 , On } , π ' = { 01 ' , o ...
Page 2185
... seen from ( i ) that ( ii ) S * ( ƒ ) = [ ƒ ( \ ) A ( dX ) , ƒ € C ( A ) . Λ It will now be shown that A is a spectral measure in X * . By placing ƒ = 1 in ( ii ) , it is seen that I * = A ( A ) and , since A ( S ) is additive in 8 ...
... seen from ( i ) that ( ii ) S * ( ƒ ) = [ ƒ ( \ ) A ( dX ) , ƒ € C ( A ) . Λ It will now be shown that A is a spectral measure in X * . By placing ƒ = 1 in ( ii ) , it is seen that I * = A ( A ) and , since A ( S ) is additive in 8 ...
Contents
SPECTRAL OPERATORS | 1924 |
The Resolvent of a Spectral Operator | 1935 |
An Operational Calculus for Bounded Spectral | 1941 |
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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 linear operator bounded operator closed operator commuting compact complex numbers complex plane converges Corollary countably additive Definition denote dense differential operator Doklady Akad domain E₁ eigenvalues elements equation exists finite number follows from Lemma follows from Theorem 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