Linear Operators: Spectral operators |
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Page 2068
... contains all inverses . There are a number of immediate corollaries to this theorem of Wiener which will become apparent from the following lemma . 6 LEMMA . Let I be an ideal in the subalgebra A。 of A and let A ( J ) be the algebra of ...
... contains all inverses . There are a number of immediate corollaries to this theorem of Wiener which will become apparent from the following lemma . 6 LEMMA . Let I be an ideal in the subalgebra A。 of A and let A ( J ) be the algebra of ...
Page 2069
... contains all inverses . Then Ag contains all inverses . -1 PROOF . If the operator A in Ag has an inverse in B ( 5 " ) then , by Corollary 9.6 , A - 1 is in " and the determinant 8 = det ( a ,, ) has an inverse in A. Since A。 contains ...
... contains all inverses . Then Ag contains all inverses . -1 PROOF . If the operator A in Ag has an inverse in B ( 5 " ) then , by Corollary 9.6 , A - 1 is in " and the determinant 8 = det ( a ,, ) has an inverse in A. Since A。 contains ...
Page 2159
... contains a non - trivial sub- interval of y . Q.E.D. 12 LEMMA ( G ) . If the point spectrum of the adjoint T * contains no non - trivial subarc of To , then the set of points regular relative to T is dense in To PROOF . If is not in the ...
... contains a non - trivial sub- interval of y . Q.E.D. 12 LEMMA ( G ) . If the point spectrum of the adjoint T * contains no non - trivial subarc of To , then the set of points regular relative to T is dense in To PROOF . If is not in the ...
Contents
SPECTRAL OPERATORS | 1924 |
Spectral Operators | 1925 |
Terminology and Preliminary Notions | 1928 |
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 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 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