Linear Operators: Spectral theory |
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Page 1216
... null set N ,. Hence for every λ not in the μ - null set N = U1 N , we have ( Tƒ ,, W " ) ( · , λ ) ) = λ ( ƒ ;, W ( " ) ( · , 2 ) ) . Since the vectors [ ƒ ,, Tf ; ] are dense in the graph ( " ) of T ( n ) we have a ( T ( n ) f , W ( n ) ...
... null set N ,. Hence for every λ not in the μ - null set N = U1 N , we have ( Tƒ ,, W " ) ( · , λ ) ) = λ ( ƒ ;, W ( " ) ( · , 2 ) ) . Since the vectors [ ƒ ,, Tf ; ] are dense in the graph ( " ) of T ( n ) we have a ( T ( n ) f , W ( n ) ...
Page 1343
... set of finite measure differs by a null set from the union of a sequence of measur- able sets on each of which the functions m ,, are continuous . Thus it suffices to prove that one can construct the functions aij and Pion any ...
... set of finite measure differs by a null set from the union of a sequence of measur- able sets on each of which the functions m ,, are continuous . Thus it suffices to prove that one can construct the functions aij and Pion any ...
Page 1914
... set of all functions , III.2.1 ( 102 ) Milman . ( See Krein - Milman theorem ) Minimax principle , X.4 ( 908 ) ... Null function . ( See also Null set ) criterion for , III.6.8 ( 147 ) definition , III.2.3 ( 103 ) Null set . ( See also ...
... set of all functions , III.2.1 ( 102 ) Milman . ( See Krein - Milman theorem ) Minimax principle , X.4 ( 908 ) ... Null function . ( See also Null set ) criterion for , III.6.8 ( 147 ) definition , III.2.3 ( 103 ) Null set . ( See also ...
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