Linear Operators: Spectral operators |
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Page 915
... denote the restriction of T to the orthogonal complement ( 1 ) of H ( 1 ) . Since T and T * map H ( ~ 1 ) into itself , T and T * map ( z1 ) into itself . Thus it follows immediately that T2 is normal . 2 Now select z in § ( ≈1 ) in ...
... denote the restriction of T to the orthogonal complement ( 1 ) of H ( 1 ) . Since T and T * map H ( ~ 1 ) into itself , T and T * map ( z1 ) into itself . Thus it follows immediately that T2 is normal . 2 Now select z in § ( ≈1 ) in ...
Page 1126
... denote by the letter U 。. λ Let S be a bounded operator in L2 ( C ) which commutes with each projection Uġ1EU 。. Let 1 denote the function in L2 ( C ) which is identically equal to 1. If U1SU 。( 1 ) = h ( x ) , then it is evident ...
... denote by the letter U 。. λ Let S be a bounded operator in L2 ( C ) which commutes with each projection Uġ1EU 。. Let 1 denote the function in L2 ( C ) which is identically equal to 1. If U1SU 。( 1 ) = h ( x ) , then it is evident ...
Page 1486
... denote the unit shift operator , so that ( Sf ) ( t ) = f ( t - 1 ) . Then , since the coefficients of 7 are ... denote n - dimensional unitary space . With each complex number 2 , associate a linear transformation B ( 2 ) in E ” , as ...
... denote the unit shift operator , so that ( Sf ) ( t ) = f ( t - 1 ) . Then , since the coefficients of 7 are ... denote n - dimensional unitary space . With each complex number 2 , associate a linear transformation B ( 2 ) in E ” , as ...
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 Haar measure 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 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 unique unitary vanishes vector zero