Linear Operators: Spectral operators |
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Page 1953
... PROOF . First suppose that 0. If TX is not dense then , by Corollary II.3.13 , there is an x * in X * with x * 0 and ... PROOF . The proof will be divided into two cases depending on whether the projection E ( { 0 } ) = 0 or not . First ...
... PROOF . First suppose that 0. If TX is not dense then , by Corollary II.3.13 , there is an x * in X * with x * 0 and ... PROOF . The proof will be divided into two cases depending on whether the projection E ( { 0 } ) = 0 or not . First ...
Page 2137
... PROOF . The proof of Corollary XV.3.3 may be used to prove the present lemma . Q.E.D. 3 LEMMA ( A ) . Let o be a set of complex numbers , and o ' its com- plement . If x + y = x1 + y1 , where o ( x ) , o ( x1 ) o and o ( y ) , o ( y1 ) ...
... PROOF . The proof of Corollary XV.3.3 may be used to prove the present lemma . Q.E.D. 3 LEMMA ( A ) . Let o be a set of complex numbers , and o ' its com- plement . If x + y = x1 + y1 , where o ( x ) , o ( x1 ) o and o ( y ) , o ( y1 ) ...
Page 2218
... PROOF . In view of Lemma 23 , the proof may be restricted to the case where the Boolean algebra B is complete . Let { E } be a weakly convergent generalized sequence in B and suppose that its limit E is a projection . It must be shown ...
... PROOF . In view of Lemma 23 , the proof may be restricted to the case where the Boolean algebra B is complete . Let { E } be a weakly convergent generalized sequence in B and suppose that its limit E is a projection . It must be shown ...
Contents
SPECTRAL OPERATORS | 1924 |
Introduction | 1927 |
Relations Between a Spectral Operator and Its Scalar | 1950 |
Copyright | |
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A₁ adjoint operator algebra of projections Amer arbitrary B*-algebra B₁ Boolean algebra Borel sets boundary conditions bounded linear operator bounded operator closed operator Colojoară commuting compact complex numbers complex plane contains converges Corollary countably additive Definition dense differential operator disjoint Doklady Akad E-measurable eigenvalues elements equation equivalent exists Foias follows from Theorem formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis identity inequality integral invariant inverse L₁ Lebesgue Lemma Math matrix multiplicity norm operators in Hilbert perturbation polynomial PROOF proved quasi-nilpotent resolution restriction Russian S₁ satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum strong operator topology subset subspace sufficiently type spectral operator unbounded unique vector weakly complete zero