Linear Operators: Spectral theory |
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Page 861
... exists , then T1 T - 1 . Clearly if x - 1 exists then T - 1T = T & T - 1 = I. If T1 exists in B ( X ) , then and if a = xα = T2 [ ( Tz1y ) z ] = yz , - ( T1y ) z = T - 1 ( yz ) , T1e , then az = T1z for every ze X. Also x T1a = e = T1 ...
... exists , then T1 T - 1 . Clearly if x - 1 exists then T - 1T = T & T - 1 = I. If T1 exists in B ( X ) , then and if a = xα = T2 [ ( Tz1y ) z ] = yz , - ( T1y ) z = T - 1 ( yz ) , T1e , then az = T1z for every ze X. Also x T1a = e = T1 ...
Page 1057
... exists and t > 0 ; and the integral 0 ( Vu ) exists and equals P S Ω ( π ) ei ( x , Vu ) dx = P En xn S Q ( Vy ) et ( v , u ) dy En lyn if P√gnQ ( Vy ) | y | -nev , u ) dy exists and V is a rotation of E " . Thus , to show that the ...
... exists and t > 0 ; and the integral 0 ( Vu ) exists and equals P S Ω ( π ) ei ( x , Vu ) dx = P En xn S Q ( Vy ) et ( v , u ) dy En lyn if P√gnQ ( Vy ) | y | -nev , u ) dy exists and V is a rotation of E " . Thus , to show that the ...
Page 1261
... exists a unique closed linear extension T such that if T1 is any closed linear extension of T then TCT . 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 T1 is any closed linear extension of T then TCT . T is called the closure of T. ( a ) There exists a densely defined operator with no closed linear extension . ( b ) An operator T ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero