Linear Operators, Part 2 |
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Page 2241
... subset of the domain of T ( 1 / f ) and vice versa . If x is in D ( T ( ƒ ) ) , then x = T ( 1 / ƒ ) T ( ƒ ) x , and so x is in the range R ( T ( 1 / f ) ) of T ( 1 / f ) . Thus R ( T ( 1 / ƒ ) ) = D ( T ( ƒ ) ) . The equation R ( T ( ƒ ) ...
... subset of the domain of T ( 1 / f ) and vice versa . If x is in D ( T ( ƒ ) ) , then x = T ( 1 / ƒ ) T ( ƒ ) x , and so x is in the range R ( T ( 1 / f ) ) of T ( 1 / f ) . Thus R ( T ( 1 / ƒ ) ) = D ( T ( ƒ ) ) . The equation R ( T ( ƒ ) ...
Page 2256
... subset σ of o ( T ) which is open in the relative topology of o ( T ) . It follows that the set 7 ( σ ) = { z | z - 1 e σ } is a compact subset of o ( R ) , open in the relative topology of σ ( R ) . Thus each point in o ( R ) different ...
... subset σ of o ( T ) which is open in the relative topology of o ( T ) . It follows that the set 7 ( σ ) = { z | z - 1 e σ } is a compact subset of o ( R ) , open in the relative topology of σ ( R ) . Thus each point in o ( R ) different ...
Page 2257
... subsets of o ( R ) which do not contain 0. Since by assumption E ( o ; T ) ] is uniformly bounded for σ in K , since each compact open subset σ of o ( R ) is either a compact open subset of σ ( R ) not con- taining zero or the ...
... subsets of o ( R ) which do not contain 0. Since by assumption E ( o ; T ) ] is uniformly bounded for σ in K , since each compact open subset σ of o ( R ) is either a compact open subset of σ ( R ) not con- taining zero or the ...
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