Linear Operators, Part 2 |
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Page 2203
... topology in A. To see this , note that sets of the form { || ( T - 1A ) ( A ) | < a } , with A in ( B ) form , by definition , a subbasis for the topology of A , and since B generates ( B ) , the sets σ ( E ) form a subbasis for the ...
... topology in A. To see this , note that sets of the form { || ( T - 1A ) ( A ) | < a } , with A in ( B ) form , by definition , a subbasis for the topology of A , and since B generates ( B ) , the sets σ ( E ) form a subbasis for the ...
Page 2278
... topology by the manifolds { N ( x * ) | x * € A } . The next lemma establishes the required continuity properties . 23 LEMMA . If E * , F * € C * and F * ≤ E * , then m ( F * ) ≤ m ( E * ) . If { E * } ≤ C * and E * = \ / \ / E * = C ...
... topology by the manifolds { N ( x * ) | x * € A } . The next lemma establishes the required continuity properties . 23 LEMMA . If E * , F * € C * and F * ≤ E * , then m ( F * ) ≤ m ( E * ) . If { E * } ≤ C * and E * = \ / \ / E * = C ...
Page 2319
... topology of A ( n ) ( J ) . = PROOF . The series 1 E ( A ; T ) f certainly converges uncondi- tionally in the topology of L2 ( J ) . On the other hand , so does the series T ( † E ( \ , ; T ) ƒ ) = Σ E ( \ , ; T ) ( Tƒ ) i = 1 1 = 1 ...
... topology of A ( n ) ( J ) . = PROOF . The series 1 E ( A ; T ) f certainly converges uncondi- tionally in the topology of L2 ( J ) . On the other hand , so does the series T ( † E ( \ , ; T ) ƒ ) = Σ E ( \ , ; T ) ( Tƒ ) i = 1 1 = 1 ...
Contents
SPECTRAL OPERATORS | 1924 |
The Canonical Reduction of a Spectral Operator | 1939 |
Bounded Spectral Operators in Hilbert Space | 1947 |
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 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