Linear Operators: Spectral theory |
From inside the book
Results 1-3 of 84
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 ( M ) | ≤ | x | this homo- morphism is continuous . 2 LEMMA . Let μ 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 ( M ) | ≤ | x | this homo- morphism is continuous . 2 LEMMA . Let μ be a non - zero ...
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 = = Let z be the element in C ( o ) with z ( 2 ) = 2 , 2 eo , and let Xo 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 = = Let z be the element in C ( o ) with z ( 2 ) = 2 , 2 eo , and let Xo 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 . 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 |
Copyright | |
52 other sections not shown
Other editions - View all
Common terms and phrases
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 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 linear linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero