Linear Operators: Spectral theory |
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Page 868
... complex number ( M ) such that x + M = x ( M ) e + M . This mapping x → x ( M ) of X into the field Ø of complex numbers is clearly a homomorphism . Since | x ( M ) | ≤ | x | this homo- morphism is continuous . 2 LEMMA . Let u be a ...
... complex number ( M ) such that x + M = x ( M ) e + M . This mapping x → x ( M ) of X into the field Ø of complex numbers is clearly a homomorphism . Since | x ( M ) | ≤ | x | this homo- morphism is continuous . 2 LEMMA . Let u be a ...
Page 1156
... complex numbers . Let f be the function of the complex variable z defined by = - a ∞ n = 1 f ( z ) = ∞ - Σα_2 " , n = 0 2 > 1 , 21 . Then a complex number t of modulus 1 is outside 1156 XI.11.9 XI . MISCELLANEOUS APPLICATIONS.
... complex numbers . Let f be the function of the complex variable z defined by = - a ∞ n = 1 f ( z ) = ∞ - Σα_2 " , n = 0 2 > 1 , 21 . Then a complex number t of modulus 1 is outside 1156 XI.11.9 XI . MISCELLANEOUS APPLICATIONS.
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
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ 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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero