Linear Operators, Part 2 |
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Page 1930
... field of sets in the complex plane , then a spectral measure E on Σ is called a resolution of the identity ( or a spectral resolution ) for the operator T if δε Σ . E ( S ) T = TE ( S ) , σ ( T¿ ) ≤ 8 , Here we have used , and shall ...
... field of sets in the complex plane , then a spectral measure E on Σ is called a resolution of the identity ( or a spectral resolution ) for the operator T if δε Σ . E ( S ) T = TE ( S ) , σ ( T¿ ) ≤ 8 , Here we have used , and shall ...
Page 2143
... field ( T ) is not necessarily a σ - field , it is natural to ask whether or not the spectral measure E may be extended to the o - field generated by ( T ) . The following definition and theorems are concerned with this question . 12 ...
... field ( T ) is not necessarily a σ - field , it is natural to ask whether or not the spectral measure E may be extended to the o - field generated by ( T ) . The following definition and theorems are concerned with this question . 12 ...
Page 2145
... field other than the field of Borel sets . Such operators are described as follows . 1 DEFINITION . Let Σ be a field of sets in the complex plane and let T be a linear operator in the complex B - space X. Then a spectral measure E on is ...
... field other than the field of Borel sets . Such operators are described as follows . 1 DEFINITION . Let Σ be a field of sets in the complex plane and let T be a linear operator in the complex B - space X. Then a spectral measure E on is ...
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