Linear Operators, Part 2 |
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Page 2127
... finite dimensional . In this case it follows from the work of Naimark [ 12 ] and Levin [ 1 ] that L is similar to a self adjoint operator on M ; hence L is a spectral operator . ( See Chapter XX where results of this character will be ...
... finite dimensional . In this case it follows from the work of Naimark [ 12 ] and Levin [ 1 ] that L is similar to a self adjoint operator on M ; hence L is a spectral operator . ( See Chapter XX where results of this character will be ...
Page 2318
... finite dimensional range for all sufficiently large m , and hence , a fortiori , E has a finite dimensional range . Q.E.D. ∞0 → 11 THEOREM . Let T be the unbounded operator in Hilbert space defined by the formal differential operator ...
... finite dimensional range for all sufficiently large m , and hence , a fortiori , E has a finite dimensional range . Q.E.D. ∞0 → 11 THEOREM . Let T be the unbounded operator in Hilbert space defined by the formal differential operator ...
Page 2360
... finite dimensional ranges . Since an operator with a finite dimensional range is clearly compact , T- is compact by Lemma VI.5.3 . This completes our proof if v > 0 . If v = 0 , we argue as follows . By ( iii ) , B ( 44 ) = { Σ ( R { μ ...
... finite dimensional ranges . Since an operator with a finite dimensional range is clearly compact , T- is compact by Lemma VI.5.3 . This completes our proof if v > 0 . If v = 0 , we argue as follows . By ( iii ) , B ( 44 ) = { Σ ( R { μ ...
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