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 of complex numbers is clearly a homomorphism . Since | x ( M ) | ≤ | x | this homo- morphism is continuous . 2 LEMMA . Let u be a non ...
... complex number ( M ) such that x + M = x ( M ) e + M . This mapping x → x ( M ) of X into the field of complex numbers is clearly a homomorphism . Since | x ( M ) | ≤ | x | this homo- morphism is continuous . 2 LEMMA . Let u be a non ...
Page 1156
... complex numbers . Let f be the function of the complex variable z defined by ∞ Σanz- " , │≈ ] > 1 , n = 1 f ( z ) = = ∞ Σαη n = 0 zn , | ≈ | < 1 . Then a complex number t of modulus 1 is outside 1156 XI.11.9 XI . MISCELLANEOUS ...
... complex numbers . Let f be the function of the complex variable z defined by ∞ Σanz- " , │≈ ] > 1 , n = 1 f ( z ) = = ∞ Σαη n = 0 zn , | ≈ | < 1 . Then a complex number t of modulus 1 is outside 1156 XI.11.9 XI . MISCELLANEOUS ...
Page 1157
Nelson Dunford, Jacob T. Schwartz. Then a complex number t of modulus 1 is outside o ( ) if and only if there exists a function g which is analytic in a neighborhood of t and is such that g ( z ) = f ( x ) for all z in this neighborhood ...
Nelson Dunford, Jacob T. Schwartz. Then a complex number t of modulus 1 is outside o ( ) if and only if there exists a function g which is analytic in a neighborhood of t and is such that g ( z ) = f ( x ) for all z in this neighborhood ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero