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 mo and hence , from Theorem 16 and Lemma 3.1 ( d ) , that there is a number a with ( h * f * q ) ( x ) = x ...
... containing the remainder of o ( f * q ) . It follows from Lemma 12 that the set ( h * f * q ) contains at most the single point mo and hence , from Theorem 16 and Lemma 3.1 ( d ) , that there is a number a with ( h * f * q ) ( x ) = x ...
Page 996
... contains no interior point of o ( q ) . 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 ( q ) . 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 ( T2 ) is a linear space , R ( T2 ) contains an element orthogonal to R ...
... 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 ( T2 ) is a linear space , R ( T2 ) contains an element orthogonal to R ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero