Linear Operators: Spectral theory |
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Page 995
... containing the remainder of o ( f * q ) . It follows from Lemma 12 that the set ( h * f * q ) contains at most the single point m 。 and hence , from Theorem 16 and Lemma 3.1 ( d ) , that there is a number a with ( h * f * q ) ( x ) a ...
... containing the remainder of o ( f * q ) . It follows from Lemma 12 that the set ( h * f * q ) contains at most the single point m 。 and hence , from Theorem 16 and Lemma 3.1 ( d ) , that there is a number a with ( h * f * q ) ( x ) a ...
Page 996
... contains no interior point of o ( p ) . Hence o ( ƒ * q ) is a closed subset of the boundary of o ( p ) . Since ƒ * q = 0 it follows from Lemma 11 ( a ) that o ( f * q ) is not void . Thus , by hypothesis , o ( fp ) contains an isolated ...
... contains no interior point of o ( p ) . Hence o ( ƒ * q ) is a closed subset of the boundary of o ( p ) . Since ƒ * q = 0 it follows from Lemma 11 ( a ) that o ( f * q ) is not void . Thus , by hypothesis , o ( fp ) contains an isolated ...
Page 1397
... contains both D ( T ) and the null - space of T * . This readily yields a contradiction as follows : the assumption ... contains the range of T1 , then , since R ( T ) is a linear space , R ( T2 ) contains an element orthogonal to R ( T1 ) ...
... contains both D ( T ) and the null - space of T * . This readily yields a contradiction as follows : the assumption ... contains the range of T1 , then , since R ( T ) is a linear space , R ( T2 ) contains an element orthogonal to R ( T1 ) ...
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