Linear Operators, Part 2 |
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Page 902
... finite number of these neighborhoods cover the set on of those complex numbers λ with | 2 | ≤ M and whose distance from σ ( T ) is at least 1 / n . Thus E ( f1 ( 8 ) ) = 0. Since E is countably additive E ( f - 1 ( p ( T ) ) ) = 0 ...
... finite number of these neighborhoods cover the set on of those complex numbers λ with | 2 | ≤ M and whose distance from σ ( T ) is at least 1 / n . Thus E ( f1 ( 8 ) ) = 0. Since E is countably additive E ( f - 1 ( p ( T ) ) ) = 0 ...
Page 1092
... finite number N of non - zero eigenvalues , we write λ ( T ) = 0 , n > N ) . Then , for each positive integer m ( a ) | 11 ( T ) . · λm ( T ) | ≤ │μ2 ( T ) . . . μm ( T ) | ; m ... m ( b ) Σ2 , ( T ) | P ≤ Σ \ μ , ( T ) | P ; j = 1 ...
... finite number N of non - zero eigenvalues , we write λ ( T ) = 0 , n > N ) . Then , for each positive integer m ( a ) | 11 ( T ) . · λm ( T ) | ≤ │μ2 ( T ) . . . μm ( T ) | ; m ... m ( b ) Σ2 , ( T ) | P ≤ Σ \ μ , ( T ) | P ; j = 1 ...
Page 1913
... finite number of finite measure spaces , III.11.3 ( 186 ) of finite number of o - finite meas- ure spaces , ( 188 ) of infinite number of finite meas- ure spaces , III.11.21 ( 205 ) o - finite , III.5.7 ( 136 ) Metric ( s ) , I.6.1 ( 18 ) ...
... finite number of finite measure spaces , III.11.3 ( 186 ) of finite number of o - finite meas- ure spaces , ( 188 ) of infinite number of finite meas- ure spaces , III.11.21 ( 205 ) o - finite , III.5.7 ( 136 ) Metric ( s ) , I.6.1 ( 18 ) ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers continuous function converges Corollary deficiency indices Definition denote dense eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero