Linear Operators: Spectral theory |
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Page 868
... complex number ( M ) such that x + M x ( M ) e + M . This mapping → x ( M ) of X into the field of complex numbers is clearly a homomorphism . Since | x ( M ) | ≤ | x | this homo- morphism is continuous . μ 2 LEMMA . Let μ be a non ...
... complex number ( M ) such that x + M x ( M ) e + M . This mapping → x ( M ) of X into the field of complex numbers is clearly a homomorphism . Since | x ( M ) | ≤ | x | this homo- morphism is continuous . μ 2 LEMMA . Let μ be a non ...
Page 1156
... complex numbers . Let f be the function of the complex variable z defined by 10 THEOREM . Suppose that = { αn } , - ∞ Σαπη , n = 1 f ( z ) = ∞ ≈ > 1 , -- - Σα3 " , 2 < 1 . n n = 0 Then a complex number t of modulus 1 is outside 1156 ...
... complex numbers . Let f be the function of the complex variable z defined by 10 THEOREM . Suppose that = { αn } , - ∞ Σαπη , n = 1 f ( z ) = ∞ ≈ > 1 , -- - Σα3 " , 2 < 1 . n n = 0 Then a complex number t of modulus 1 is outside 1156 ...
Page 1157
Nelson Dunford, Jacob T. Schwartz. Then a complex number t of modulus 1 is outside o ( p ) if and only if there exists a function g which is analytic in a neighborhood of t and is such that g ( x ) = f ( z ) for all z in this ...
Nelson Dunford, Jacob T. Schwartz. Then a complex number t of modulus 1 is outside o ( p ) if and only if there exists a function g which is analytic in a neighborhood of t and is such that g ( x ) = f ( z ) for all z in this ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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Common terms and phrases
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 Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm open set 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 unique unitary vanishes vector zero