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 872
... complex plane whose complement is connected . Let C ( a ) be the B - algebra of all continuous complex functions defined on σ with . norm = If = sup f ( 2 ) . λεσ X Let z be the element in C ( o ) with ( λ ) = λ , λ € σ , and let o be ...
... complex plane whose complement is connected . Let C ( a ) be the B - algebra of all continuous complex functions defined on σ with . norm = If = sup f ( 2 ) . λεσ X Let z be the element in C ( o ) with ( λ ) = λ , λ € σ , and let o be ...
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 H. 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 H. The symbol ( x , y ) will be used for the scalar product of the vectors x and y in H. By definition ...
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