Linear Operators, Part 2 |
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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 . μ 2 LEMMA . Let u be a non - zero ...
... 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 . μ 2 LEMMA . Let u be a non - zero ...
Page 872
... complex plane whose complement is connected . Let C ( o ) be the B - algebra of all continuous complex functions defined on σ with norm Ifl = sup [ f ( 2 ) . λεσ = Let z be the element in C ( o ) with 2 ( λ ) = λ , λ e σ , and let Xo be ...
... complex plane whose complement is connected . Let C ( o ) be the B - algebra of all continuous complex functions defined on σ with norm Ifl = sup [ f ( 2 ) . λεσ = Let z be the element in C ( o ) with 2 ( λ ) = λ , λ e σ , and let Xo be ...
Page 1156
... complex numbers of unit modulus . Using the second of these realizations for R we have [ n , λ ] = 2 " where n ɛ R ... complex numbers . Let f be the function of the complex variable z defined by - anz - n , | ≈ > 1 , f ( z ) = ∞ -Σα ...
... complex numbers of unit modulus . Using the second of these realizations for R we have [ n , λ ] = 2 " where n ɛ R ... complex numbers . Let f be the function of the complex variable z defined by - anz - n , | ≈ > 1 , f ( z ) = ∞ -Σα ...
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 domain 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(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma 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