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 1951
... ideal J in B ( X ) . Then every projection E ( o ) with 0 ₫ ō belongs to J. If I is closed , then S and N also belong to J. PROOF . Let 0 ō and let T , TE ( o ) | E ( o ) X , the restriction of T to the invariant subspace E ( o ) X ...
... ideal J in B ( X ) . Then every projection E ( o ) with 0 ₫ ō belongs to J. If I is closed , then S and N also belong to J. PROOF . Let 0 ō and let T , TE ( o ) | E ( o ) X , the restriction of T to the invariant subspace E ( o ) X ...
Page 1952
... ideal , and this ideal is closed if { x } is bounded . Thus the corollary follows directly from Theorem 2. Q.E.D. 10 THEOREM . Let A be a bounded linear operator in X. Then AT = 0 if and only if AN = 0 and AE ( { 0 } ' ) = 0. Similarly ...
... ideal , and this ideal is closed if { x } is bounded . Thus the corollary follows directly from Theorem 2. Q.E.D. 10 THEOREM . Let A be a bounded linear operator in X. Then AT = 0 if and only if AN = 0 and AE ( { 0 } ' ) = 0. Similarly ...
Page 2264
... ideal if E , F D implies E v Fe D and GE , EE D , implies Ge D. The ideal D is dense if every element of B is a union of elements of D. A o - ideal is an ideal closed under countable unions . 2 LEMMA . Let D be a dense ideal in 2264 ...
... ideal if E , F D implies E v Fe D and GE , EE D , implies Ge D. The ideal D is dense if every element of B is a union of elements of D. A o - ideal is an ideal closed under countable unions . 2 LEMMA . Let D be a dense ideal in 2264 ...
Contents
SPECTRAL OPERATORS | 1924 |
Relations Between a Spectral Operator and Its Scalar | 1950 |
The Algebras and | 1967 |
Copyright | |
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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