Linear Operators, Part 2 |
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Page 1967
... element in X with an inverse in Y has this inverse also in X. - e = ( ee * ) * = e = PROOF . We first show that e ... element z = xx * is in X. If z - 1 is in X , then e = xx * z - 1 and so x has its inverse x * z - 1 in X. This reduces ...
... element in X with an inverse in Y has this inverse also in X. - e = ( ee * ) * = e = PROOF . We first show that e ... element z = xx * is in X. If z - 1 is in X , then e = xx * z - 1 and so x has its inverse x * z - 1 in X. This reduces ...
Page 2108
... element a Є A is real ( respectively , positive ) in case its spectrum o ( a ) is contained in R ( respectively , [ 0 , + ∞ ) ) . It can be proved that an a € A is a real scalar element if and only if there exists an ordering of A with ...
... element a Є A is real ( respectively , positive ) in case its spectrum o ( a ) is contained in R ( respectively , [ 0 , + ∞ ) ) . It can be proved that an a € A is a real scalar element if and only if there exists an ordering of A with ...
Page 2215
... element of B. 1 PROOF . It is clear that every element in the weakly closed operator algebra generated by B commutes with every element of B. To prove the converse , let A commute with every element of B , and let B1 be the strong ...
... element of B. 1 PROOF . It is clear that every element in the weakly closed operator algebra generated by B commutes with every element of B. To prove the converse , let A commute with every element of B , and let B1 be the strong ...
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