Linear Operators: Spectral theory |
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Page 861
... exists , then TT - 1 . Clearly if x - 1 exists then T - 1T = TT - 1 = I. If T exists in B ( X ) , then T2 [ ( Tz1y ) z ] = yz , ( T = 1y ) z = T21 ( yz ) , and if a = Thus x xα = T1e , then az = T1z for every ≈ Є X. Also ( Te ) x = ax ...
... exists , then TT - 1 . Clearly if x - 1 exists then T - 1T = TT - 1 = I. If T exists in B ( X ) , then T2 [ ( Tz1y ) z ] = yz , ( T = 1y ) z = T21 ( yz ) , and if a = Thus x xα = T1e , then az = T1z for every ≈ Є X. Also ( Te ) x = ax ...
Page 1057
... exists for each u . By Lemma 2 , the integral 0 ( tu ) exists if 0 ( u ) exists and t > 0 ; and the integral ( Vu ) exists and equals Ω ( α ) P ei ( x , Vu ) dx Гр = En xn 水 Ω ( Vy ) ei ( v , u ) dy En y " if Pæn Q ( Vy ) | y | - " ei ...
... exists for each u . By Lemma 2 , the integral 0 ( tu ) exists if 0 ( u ) exists and t > 0 ; and the integral ( Vu ) exists and equals Ω ( α ) P ei ( x , Vu ) dx Гр = En xn 水 Ω ( Vy ) ei ( v , u ) dy En y " if Pæn Q ( Vy ) | y | - " ei ...
Page 1261
... exists a unique closed linear extension T such that if T , is any closed linear extension of T then TCT1 . T is called the closure of T. ( a ) There exists a densely defined operator with no closed linear extension . ( b ) An operator T ...
... exists a unique closed linear extension T such that if T , is any closed linear extension of T then TCT1 . T is called the closure of T. ( a ) There exists a densely defined operator with no closed linear extension . ( b ) An operator T ...
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 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