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 872
... complex plane whose complement is connected . Let C ( o ) be the B - algebra of all continuous complex functions defined on σ with norm In particular 20 - = = a ( z ) ( Lemma 4 )げ= sup f ( 2 ) . λεσ Let z be the element in C ( o ) with ...
... complex plane whose complement is connected . Let C ( o ) be the B - algebra of all continuous complex functions defined on σ with norm In particular 20 - = = a ( z ) ( Lemma 4 )げ= sup f ( 2 ) . λεσ Let z be the element in C ( o ) with ...
Page 887
... complex numbers . Throughout the chapter the symbol T * will be used for the Hilbert space adjoint of the operator T in Hilbert space . The symbol ( x , y ) will be used for the scalar product of the vectors x and y in H. By definition ...
... complex numbers . Throughout the chapter the symbol T * will be used for the Hilbert space adjoint of the operator T in Hilbert space . The symbol ( x , y ) will be used for the scalar product of the vectors x and y in H. By definition ...
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