Linear Operators: Spectral theory |
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Page 1188
... defined by the identity ( Tx , y ) = ( x , T * y ) . We shall need to use the notion of the Hilbert space adjoint of an operator which is not necessarily bounded and this concept is formulated in the following definition . DEFINITION ...
... defined by the identity ( Tx , y ) = ( x , T * y ) . We shall need to use the notion of the Hilbert space adjoint of an operator which is not necessarily bounded and this concept is formulated in the following definition . DEFINITION ...
Page 1196
... defined in Definition 1.1 or as in Definition VII.9.6 . This is the case , as will be shown in Corollary 8 below , so that the symbol f ( T ) for a polynomial ƒ is unambiguously defined . 6 THEOREM . Let E be the resolution of the ...
... defined in Definition 1.1 or as in Definition VII.9.6 . This is the case , as will be shown in Corollary 8 below , so that the symbol f ( T ) for a polynomial ƒ is unambiguously defined . 6 THEOREM . Let E be the resolution of the ...
Page 1647
... DEFINITION . Let be a formal partial differential operator defined in an open subset I of E " , and with coefficients in C °° ( I ) . Let F be a distribution in I. Then TF will denote the distribution defined by the equation ( tF ) ( q ) ...
... DEFINITION . Let be a formal partial differential operator defined in an open subset I of E " , and with coefficients in C °° ( I ) . Let F be a distribution in I. Then TF will denote the distribution defined by the equation ( tF ) ( q ) ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero